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A New Look Towards the Principles of Motion - 1
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Arindam Banerjee
May 26, 2019, 11:23:15 PM5/26/19
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Section 1
Linear Motion, Momentum, Force, Energy, Internal Force Engines, and the design of Interstellar Spacecraft
Inertial Frame of Reference
We must first consider an “inertial frame of reference”. For most practical terrestrial purposes any point notionally fixed on the Earth is valid as an inertial frame of reference. Speaking more generally, an inertial frame of reference is a specified point, line, area or volume such that all points that are fixed on that particular frame of reference do not move with respect to each other; and such an inertial frame of reference is not accelerating away from another such inertial frame of reference. Two points fixed on the Earth do not move with respect to each other (unless there is an earthquake or continental drift in which case a re-mapping is required). So the fixed points on the Earth constitute a frame of reference, roughly referred to as “ground”.
What velocity a particular frame of reference have with other such frames of reference may or may not be known. Thus we can never know the velocity of the Earth, at any point on its surface, with respect to the fixed, static, all-pervading and solid medium which has been defined as ether; through which all electromagnetic waves have to travel. (For while we may find out the orbital speed around our Sun, we cannot know the speed of the solar system in the galaxy, the speed of the galaxy itself, or the speed of the intergalactic cluster our galaxy belongs to, as they all progress in space.) On the other hand, we can know the velocity of an airplane or clouds with respect to the ground (the inertial reference frame for the aircraft), with aid of such a tool as radar. But for the passengers and crew in the airplane, who are far away from the ground, the inertial reference frame relevant to them is their moving airplane itself. Thus, no matter what the speed of the airplane with respect to its inertial reference frame – the ground directly below it at any time – the distance between the passenger and say the door of the airplane will remain constant. And if the passenger is still, his speed with respect to the airplane (his inertial reference system) will be zero, while with respect to the ground directly below him, his speed will be the same as that of the airplane.
One-dimensional space
The following equations that will be derived, are essentially in one-dimensional space. Multi-dimensionality, involving vector algebra, is essentially for practical engineering issues – the consideration of multi-dimensional space will make the understanding of the fundamental equations of motion more complex, without adding any further necessary illumination at this preliminary stage. After all, to go to the stars, all we have to do is to point the nose of our starship to the required star, and just go there! That would be a very uni-dimensional path, indeed! So for the purpose of understanding, so far as this present article is concerned, we limit our inertial frame of reference not over the whole curved planet, but the small part of it around us that is flat or linear (as is the case for most of our day to day life, unless we happen to be astronauts in far outer space, in which case our initial frame of reference is the point that is the centre of the Earth). In any case, making the mathematics simple should make this article understandable to anyone of acute intelligence above 13 years of age, and with some knowledge of algebra.
Fundamental Equations of Motion
The fundamental equations of motion, upon which most engineering are ultimately based are very simple. I will show how simple they are, by deriving them from the first principles, below.
1. s = v_avg*t. s is the distance covered over the time t by a body moving with average velocity v_avg (average rate of change of distance with time) along the distance. Thus if a body is moving with an average speed of 10 meters per second in a straight line, in 5 seconds it moves 50 meters away from the fixed point it was at 5 seconds ago.
2. v = u + a*t. v is the final velocity of the body in motion, u is the initial velocity, t is the time period over which the body changes from velocity u to velocity v without change in direction, and a is the average acceleration (rate of change of velocity with time) over the time period along the straight line. Thus if a body moves with a speed of 10 meters per second past the reference point, accelerates continuously at 2 meters per second per second on the average (like a car, say), after that, then after 10 seconds of undergoing such acceleration it will have the velocity of 10+2*10 = 30 meters per second.
These two equations written above are the most basic. It should be clear that they do not deal with any turning motion – merely motion in a straight line.
The following two equations are derived from the above.
3. s = u*t + 0.5*a*t*t (the variables have been defined earlier, but here the acceleration value is constant throughout the time period t.)
Derivation:
v_avg = (u+v)/2, if the acceleration is uniform over the time t.
s = v_avg*t = ((u+v)/2)*t = ((2*u+ at)/2)*t = u*t + 0.5*a*t*t.
If u=0, or the body starts from rest, then s = 0.5*at*t.
4. v*v = u*u + 2*a*s (the variables are as defined earlier, and here too a is constant over the time interval between u and v).
Derivation:
v*v = (u + at)*(u + at) = u*u + 2uat + a*a*t*t = u*u + at(2u +at)
= u*u +at(u + v) = u*u + at*2*v_avg = u*u + 2*a*t*v_avg
= u*u + 2*a*s
The above four equations are the most fundamental to describe linear motion in what is known as the Newtonian frame of reference. They have been know for centuries.
I would like to point out their extreme simplicity and elegance to the reader.
Momentum and Rocket Science
We now come to the very important concept of momentum of a body. By body we mean an amount of matter contained within a closed geometry. A closed geometry is like, say, a box or a ball.
Now, what is matter? Matter is anything that can exert force, when linearly accelerated, upon other matter; or have its smallest components vibrate, as a consequence of travelling electromagnetic waves. It requires force for acceleration. (We will talk about force, later.) All matter is influenced by the force of gravity, which is always present on, near and within the Earth. From our Earthly perspective, matter is whatever on Earth that has weight.
The quantity of matter contained within the body, is called its mass. Units of mass are grams, ounces, tons, kilograms, etc.
The momentum of a body is simply the product of its mass and velocity. Thus M = m * v, where M is the momentum, m is the mass and v is the velocity of the mass. Just as the velocity is relative to an inertial reference system, the momentum is also thus relative to the inertial reference system.
The above equation is simple. What is slightly more difficult to understand is the concept of the law of conservation of momentum. The definition of the law of conservation of momentum is simple, though. It says that the momentum of a system of bodies in motion is always constant. So M = m(1)*v(1) + m(2)*v(2) +…. +m(n)*v(n), where the m(1), v(1) terms refer to the mass and velocity of the first body, and so on. In our simple one-dimension world, the v() terms could have only positive or negative signs, meaning the masses going one way, or the opposite way.
Fundamentals of Rocket Science (in terms of purely momentum) expressed in (relatively) simple mathematics
It is from this simple law of conservation of momentum, that the fundamentals rocket science may be discovered. Suppose the question is, what is the size of the rocket which will take a piece of mass (in outer space, away from large bodies such as the Sun or the Earth) to the speed of light, and bring it back to its rest speed? The law of conservation of momentum will let us find that out!
Note: Please leave out this section for now, if you feel somewhat challenged by the mathematics!
Now by the law of conservation of linear momentum, when a body of mass m is ejected from a body of resting mass M (that is, its velocity is zero with respect to the inertial reference frame) in some given inertial reference frame, then the mass M gets a velocity V of magnitude m*v/(M-m) and in the direction opposite to v, where v is the velocity of the ejected mass m. For M*0=0=mv – (M-m)V, or V/v=m/(M-m).
Using this basic approach, let us have:
a large rocket with a payload pointing at a star to which it must go;
the rocket is at rest, in the inertial reference frame, call it the first inertial reference frame;
an engine which ejects bodies (gases, whatever) at a constant velocity v in the opposite direction of travel of the rocket;
the bodies ejected decrease in mass at a proportionate rate, say at r percent of the existing rocket mass with every ejection.
Then we have for the first ejection: m = M*r, and so m*v = M*r*v,
and M*r*v= (1-r)M*V from the law of conservation of momentum, or V=r*v/(1-r); for ejected mass m = M*r, and (1-r)*M is the balance mass, for M*(1-r) + r*M = M.
Also, if v/V = X, or the number of times the speed of the ejecting mass is greater than that of the increased rocket velocity, then X = (1-r)/r. Then rX = 1-r, or r = 1/(1+X).
After this the rocket of mass M(1-r) will move at velocity V, indefinitely, if there is no impediment. The new inertial frame of reference is this rocket of mass M*(1-r) moving at speed V.
Let it now further throw out a mass r(1-r)*M out at speed v, or momentum r(1-r)*M*v, with respect to this new inertial frame of reference.
Then the mass of the rocket left will be M*(1-r) - r(1-r)*M, or M*(1-r)(1-r).
So the velocity gained by the rocket will be r(1-r)**v/(M*(1-r)(1-r)) = rv/(1-r) = V.
Then, with respect to the rocket's earlier speed, that is V, the speed of the rocket will increase by r(1-r)*M*v / M*(1-r)(1-r) or r*v/(1-r) which is the same amount as earlier, that is V. With respect to the first inertial reference frame, is speed now will be 2*V.
After two ejections, the mass of the rocket is (1-r)^2*M; we can see this way that after N ejections it will be (1-r)^N*M. And the speed of the rocket will be N*V, with respect to the first inertial reference frame.
Let us now try to calculate the size of the rocket that will take us to the speed of light (which is what we must reach if we are ever to go to the nearby stars).
Now c/V, (c is the speed of light) is the value of N, that is, the number of mass ejections required for the payload to reach light speed.
Let the mass of the rocket, which was M at zero speed with respect to the star it wanted to reach, at light speed be N. Then from the above derivation N=(1-r)^(c/v) * M.
And M, the mass of the rocket at zero speed, will be:
M = N/((1-r)^(c/V)).
Putting the value of r = 1/(1+X), the final formula for the mass M of the rocket required to send a payload N to the speed of light is:
M = N/(1-(1/(1+X)))^(c/V), where X = v/V
Now c = 3*10^8 m/sec. If V = 30 m/sec, then c/V = 10^7. If v = 3000 m/sec, then X = 100, and (1-r) = 0.99. And (c/V) factor is 10^7.
Putting these values we get M = N/(.99)^10000000, an extremely high number, which even the powerful modern calculator cannot display. To get some idea, let us use old fashioned log tables. The log of .99 is -0.0043648, so the log of (.99)^10000000 = -43648. Taking antilog of -43648, we find that the factor M/N is 10^43648, which is 1 followed by 43648 zeroes! Now, the integer Google, which is 1 followed by a 100 zeroes, is a sufficiently large mind boggling number – this one baffles belief! Can we make our rocket smaller?
Only if X is of the order of c/V, is there any chance of making a rocket of reasonable size. For this, let us see
When X=c/V, then (1-r) = (1 – 10^(-7)) or 0.9999999, Then M = N/(0.9999999)^10000000 = N/0.3678 = 2.71828 N.
So the mass expelled for the rocket to reach light speed is M – N = 2.71828N – N = 1.71828 N
Well, this will take us to light speed. It can coast at this speed for a while, say a few years. But then it has to slow down. This can be done be turning around 180 degrees, and decelerating by throwing out mass the other way to what was being done previously. So to return to rest speed (with respect to the first inertial reference frame) we have N = 2.71828 * Q, where Q is the final payload at rest in the faraway star.
So the size of the rocket to deliver the payload Q will be M = 2.71828*N or 2.71828^2 * Q = 7.389 Q. So as much as 6.389 times Q (final payload) will have to be spent as rocket fuel mass to be ejected. Not too efficient, what? Besides, what with the trail of matter following the rocket’s linear path, later trips could well be highly jeopardized or well-nigh impossible, what with great risk of collision.
Still, this does not look too bad! Is there a catch anywhere? Unfortunately, there is one.
When X = 10^7, and V=30m/sec, then the speed of the exhaust mass v =V*X has to be 3*10^8 m/second, which is the speed of light! To throw out mass at light speed, is beyond the scope of modern science and technology. But only if we throw out mass at that speed, can we reach light speed for the rocket. But how to accelerate any mass to light speed in the first place!
Thus we find, mathematically, that it is impossible to go to the nearest star, or reach light and faster than light speed, with rocket technology. It is simply out of the question! Either we make a rocket of impossibly large size, or we throw out mass at light speed. So, are we really stuck for ever on Planet Earth? Let us see! Having understood the concept of momentum, we shall now try to understand the concept of force, in the sense of physics.
Force
The basic definition of force is given by Leonardo da Vinci, one of the greatest geniuses of all time, in one of this notebooks.
“Force I define as a spiritual power, incorporeal and invisible, which with brief life is produced in those bodies which as the result of accidental violence are brought out of their natural state and condition. I have said spiritual because in this force there is an active, incorporeal life; and I call it invisible because the body in which it is created does not increase in weight or in size; and of brief duration because it desires perpetually to subdue its cause, and when this is subdued it kills itself.”
Thus Leonardo defines force in metaphysical terms. Force emanates from the realm of the spirit. It is the constant interaction between the Divine and the material – and its key characteristic is life. The word Leonardo used for force, “spiritual power”, may for clarity better be replaced as “spiritual manifestation”; as “power” has a meaning different from “force” in the science of dynamics. (Specifically, power is rate of usage of energy, and energy as we will see can be measured in terms of force multiplied by distance.)
Force is essentially something mysterious. It is the ultimate show of life whenever life forms are involved, or otherwise there is something Divine, Purely Random or Divinely Sanctioned Random quality about force, when non-living objects are involved. Just as mass is what is moved by force, force is what moves mass.
On earth, we have to continuously experience the force of gravity – one of the most fundamental forces of Nature. This means, that if a body is let go near the surface of the earth, it will accelerate constantly towards the surface of the earth with an acceleration value of g or roughly 9.8 meters per second per second.
However, Newtonian mechanics is not based upon the necessity of the Earth. It is abstract – when applied to Earth it takes in the gravitational pull of the Earth. So when we talk of Newton’s laws of motion, we do not have to take the Earth into account as a must. They apply everywhere.
The concept of mass comes into Newton’s laws. Mass is simply a lump of matter, which could exist as solid, liquid, gas or in a charged state known as plasma. Matter is what is moved, or can be moved, with force. A particular piece of solid matter is taken as a certain standard of mass. Such standards known are the kilogram, pound, gram, ton, etc.
We ignore Newton’s first and third laws, for now, and concentrate upon the second law. The second law defines force mathematically in terms of the concepts of momentum.
The second law of Newton is stated as: Rate of change of momentum is equal to the applied force and takes place opposite to the direction in which the force is applied. So if the mass of the body is M, and the initial velocity is u, then its initial momentum is M*u. If now a force acts upon it, causing acceleration, then after a time t over which this force is acting, the velocity of this body is v. And its momentum is thus m*v. The change of momentum is thus m*u – m*v, or m*(u-v). The rate of change of momentum is m*(u-v) divided by the time over which this change took place, which is t. So the rate of change of momentum is m*(u-v)/t. And by the second law, this is equal to the applied force F, which has been acting for t seconds upon the mass.
So the equation is F = -1*m*(u-v)*t, the minus sign coming because the force is opposite to the direction of rate of change of momentum, thus
F = m*(v-u)/t = m*a. Thus the force is always in the direction of the acceleration.
This is a crucial equation. To see its use, let us see (below) how this equation alone can explain the movement of rockets.
Force in Rocketry
As we know, a rocket works when mass is ejected out of it at high speed. It goes in the opposite direction to which mass is ejected. When it ignited, hot gas comes out, but it is only after a while that the rocket starts to rise from the Earth. That happens when the force exerted by the exiting gases are equal to the (at that time) weight of the rocket. As the gases leave the rocket, the mass of the rocket also decreases by that amount. If the gases are leaving the rocket at a constant rate of v meters per second with respect to the rocket’s end, then the momentum they exert upwards will be m(t)*(v(t)-0) or m(t)*v(t), where the inertial reference frame is the rocket itself. And force F at time t, that is, the rate of change of momentum, will be F(t) = time derivative (d/dt) of m(t)*v(t) that is
F(t) = m(t)*d(v(t))/dt + v(t)*d(m(t))/dt.
Some elementary knowledge of differential calculus is necessary to understand how the above equation was derived. I request my clever 13 year old reader, who knows algebra but has yet to study calculus, to simply accept this, for now!
m(t) or the mass coming out of the rocket at time t over a very small time interval dt approaches zero as dt (the very small interval of time) approaches zero, so the first term in the above equation tends to zero in the limit. Also, if v(t) is a constant, then the term d(v(t)/dt has to be zero. On the other hand, the term d(m(t))/dt has a finite value – it is equal to the loss of mass of the rocket over a particular time period. Thus the equation reduces to
F(t) = v(t)*d(m(t))/dt , or the velocity of the escaping gases with respect to the rocket multiplied by the rate of loss of mass of the rocket.
To give some values, if the exiting speed of gases is 1000 m per second, and mass is lost at 1 Kg per second, then the force exerted upwards will be 1000 MKS units (newtons). That will support a mass of little more than 100 Kgs (thus give it “antigravity”). If say the speed of the exiting gases is increased to 1200 m per second, with the same rate of loss of mass, then a net upward force of 200 newtons will be exerted upon the mass of 100 Kgs – thus giving it a 2 meter per second per second acceleration. So, the mass will go up with increasing speed.
All this should square with our personal experiences with rockets. In NASA launches, we see gases come out for a while before the rocket starts to rise. To begin with, the gases do not come out with such speeds as would be necessary to provide the lift. As they do so, the rocket ascends faster and faster. But, it has to keep on losing mass to do so. We have seen earlier, there is no way it can ever reach light speed, and so, rocket science can never take us much beyond the solar system.
Must we be tied to Earth and the Solar System at most?
Modern physics cannot have it otherwise! We have found out that it is impossible to construct a rocket to reach light speed. Apart from this, there are certain theories relating to relativity that make it impossible to reach the speed of light. Apparently, strange things happen when we are nearing light speed – mass becomes extremely massive, length becomes near zero… We will talk about these concepts later.
If we think a bit more deeply, we will find that it is actually our pre-occupation with rocket science that is preventing us from reaching the nearby stars. Cannot there be any other, or more advanced science, that can overcome the issues that prevent us from reaching the stars? This was dealt with in detail in articles and a book published in Internet by the author nearly ten years ago – the name of the book is “To the Stars!” and is freely available (since around 2000 AD) in the website www.users.bigpond.com/adda1234/index.htm The original concepts first laid out there, form the basis of what may be popularly known as anti-gravity; and these basic principles show the way to go to the stars. We will here deal with them in brief.
Internal Force (Anti-Gravity)
So how does a man in a rocket actually get off the ground? The answer is, as given earlier, when the force exerted upward by the exhaust gases are more than the rocket’s weight. The gases are created by using up the energy within the rocket.
Suppose there is a different way to create upward force, using up energy stored or generated within the rocket, of which we do not yet know anything about. We further assume that this upward force did not involve loss of matter from the rocket.
So Upward Force U > M*g, where M is the mass of our spaceship (yes, it is not a rocket ship any more) and g is the acceleration due to the gravity of our planet. Putting U = M*a, then M*a > M*g, and so the net force N that is accelerating the spaceship up is M*(a-g). As the spaceship goes further away from earth, the value of g decreases till it becomes nearly zero. When it does so, we are then accelerating with the acceleration a, now up from the initial value (a-g).
Reaching light speed
Equation 2 in Section 1 is
V = u + a*t
When v is the speed of light, and u is zero (with respect to the initial reference frame) and a is say slightly more than g or 10 m/sec/sec, then
3*10^8 = 10*t, or it takes 3*10^7 seconds to reach light speed. Which comes out to 347 days, or slightly less than one year.
Reaching the nearest star
The nearest star is a little over four light years away. So after reaching light speed, or exceeding it, the spaceship can coast for say 3 years. Then it turns around 180 degrees till it faces the Earth, and starts the engines as it did. This time this will be deceleration, and it will take another 347 days for the spaceship to come to rest at the destination star. So overall, about 5 years will be required to make a one-way trip to the nearest star!
Newton’s Third Law
Thus, if only we do not think in terms of throwing out mass at high speed, in order to accelerate a vehicle, then only can we reach light speed.
Is it possible to accelerate a vehicle without throwing out mass? We do have cars, and bicycles. These do not accelerate by throwing out mass, nor are animals needed to pull them. How do these work? These use internal force, from the internal energy from the cyclist’s muscles or the fuel needed for the internal combustion engine.
Cars and bicycles do use internal force, but they cannot operate without external force. The movement of the wheels upon the ground causes frictional forces to operate – the ground is pushed back by the wheels, and the ground pushes the car or bicycle forward. The more strongly the ground is pushed backed, with equal vigor the car or bicycle is pushed forward by the ground. In short, without the whole Earth (or at least a substantial part of it) to push back, there is no chance of forward motion for the car or bicycle. So if we raise the rear wheel of the bicycle using a stand, and pedal fast and furiously, we go nowhere – only the wheel spins faster and faster!
About airplanes, now, that do use internal force and do not use friction. The propeller in the airplanes push back the air, and with as much force that the air is pushed back, the air pushes the airplane forward. Similarly a jet engine sucks in air, heats it by burning fuel in it, and then pushes out the heated and polluted air with great force – usually greater than the propeller airplanes, and so jet engines cause airplanes to fly faster. No airplane can fly where there is no air (such as in outer space, where you can accelerate only if you push out part of your own mass), just as cars and bicycles cannot move if they cannot push back the earth with their wheels.
From the above we find examples of the truth that every force exerts an equal and opposite force, for all known mechanical phenomenon involving force. This has been found to be universal and so far (according to all the scientific books and papers on the physics of motion, save just one, about which we will talk about later) has no exceptions. It is a law of nature – according to all the natural philosophers.
Newton’s First Law of Motion
We now have made clear the fundamental concepts of momentum and force. There is another very important fundamental concept, inertia. Isaac Newton dealt with this concept very carefully. Inertia is defined as the property of the material body to stay still, or to move with a constant velocity, in the absence of any forces affecting the body. Every material body, it has been found, does normally exist in a state of inertia. So universal is this phenomenon, that it is deliberated upon as a law of nature, and is stated as “Every body stays in a state of rest, or in a state of uniform motion in a straight line, unless affected by an externally impressed force.”
So a car moves when the earth pushes it (this is the external force) when there is frictional contact between the wheel and the ground; the airplane moves when the air forced back by the propeller motion pushes it; the rocket moves when the gases pushed out from it, push back the rocket.
Changing Newton’s First Law
The word “externally” in the definition of the First Law of Motion has somehow managed to insinuate itself there, detracting thus from the concept of inertia. Suppose we simply delete it, and see what are the consequences. Then the First Law of Motion will be: “Every body stays in a state of rest, or in a state of uniform motion in a straight line, unless affected by an impressed force.”
This implies, that some energetic activity within the body, can give it motion without any need for friction, for pushing out mass, or pushing back mass. In other words, it behaves like a flying saucer is expected to behave – it just takes off!
Motion of a Flying Saucer, or Future Space-Ship
Let us try to get a little mathematical once again, in order to better understand the flight of the flying saucer, or space-ship. Suppose a net force F acts on the body, this force arising from the energies arising from within the body, for a time t. The mass of the ship is M. Then the acceleration a will be F/M. Since this acceleration acts over the time period t, after the time t, the velocity of the ship will be from the formula v = a *t,
V = (F/M) * t.
So after reaching the speed V from the initial zero velocity (with respect to the inertial frame of reference), if all internal activity ceases, the ship will continue to move with the speed V with respect to that original inertial frame of reference. Now this is something like firing a bullet in deep space: from zero velocity with respect to the gun, originally, the bullet keeps on going at speed V with respect to the gun, for all time to come (unless it hits a meteor or comet or planet or star).
Now after reaching the speed V, the activity same as what was done before again takes place within the ship – a Force F acts for a time t on the ship. Now the inertial frame of reference for this force, is the space-ship moving with constant speed V (this speed V is with respect to the original inertial frame of reference). Thus this force will cause a further addition of velocity, V, to the ship. With respect to the original inertial frame of reference, its speed will now be V+V=2V, after two “hits”.
Thus after N such hits, the speed of the flying saucer will be N*V. If V is 10 meters/ second, then we have seen earlier how light speed can be reached in less than one year, at this rate.
Travel to the Moon
However to go to the moon or other planets, we do not need to fly that fast, at the speed of light. Let us see how long it will take us to go to the moon, with this ship, using the laws of motion see earlier. The distance from the Earth to the Moon is 400,000 Km.
To travel half the distance or 200,000 Km, or 200,000,000 meters accelerating at 10m/sec will be from the formula S = u*t + 0.5*a*t*t, putting u=0,
200,000,000 = 0.5*10*t*t, or t*t = 40,000,000, or t = 6324seconds.
To travel the remaining 200,000 Km, by turning 180 degrees and decelerating at the same rate, will take the same time. So the total time for travel to the moon will be 12648 seconds, or around four hours. The maximum speed reached will be from the equation v = u + a*t, or v = 0 + 10*6324 = 63,240 meters per second.
The concepts of Energy, Engine and Work
Crucial to the design of flying saucers powered with Internal Force Engines, is our understanding of the concepts of energy, engine and work.
We had in the previous section sneaked in the concept of “Energy” from which the Force is generated. What is this energy?
Energy is another very fundamental concept, like mass, momentum, and force. It is the source of much talk in the world today. It’s usage (whether to get it from fossil fuels or renewable sources such as the solar, wind, tidal, geo-thermal, hydroelectric, etc.) is much debated. However few people other than practical engineers have a clear idea about the nature of energy, in the scientific sense.
Essentially Energy is defined in terms of the Work that is done by a body. As much Work is done, at least so much energy is required. I say “at least” because usually only a fraction of the energy is converted into useful work by any engine – the remaining energy is usually lost in the form of “heat”; or rather, in the dissipation of energy as heat, flowing into the universe, as it cannot be converted into work, or converted into other forms of energy.
I did mention above the word “engine” – so what is an engine? An engine is a mechanical or electro-mechanical device which converts energy (from whatever source) into work. A heat engine is one which converts heat energy into work. The earliest example of the heat engine is the steam engine. Water was heated by burning wood or coal; thus high-pressure steam was generated, and this steam was used to push a piston up and down a cylinder; and this linear movement was converted into rotary movement by mechanical contraption to rotate wheels. Now a lot of heat that was generated was not converted into work – thus the heat engine had low efficiency. An engine that uses electrical energy to produce work (an electrical motor that is) is a lot more efficient than a heat engine.
So what is Work? This is the key question. Work is measured by the following formula: Work = Force * Distance, where the Distance (an unit of length) must be aligned in the direction of the force. In other words, if a force F moves a body of mass M over a distance D, then the Work Done is F*D, or M*a*D, where F = M*a.
What work is done in carrying say a suitcase and also the the carrying person totaling mass M along a flat corridor? The person who carried it surely must think he has done some work, for his muscles tell him something! But according to our formula, no work has been done! For the force that has been acting on the suitcase, gravity, is at right angles to the direction of the motion of the suitcase. But if the person carries the suitcase up a flight of stairs, of height say D, then he has done the Work measured in terms of M*g*D, where g is the acceleration due to the Earth’s gravity, and the force acting upon the suitcase.
Back to the concept of energy, now. The energy to lift the suitcase (and the person too) came from the carrier’s muscles. To sustain the muscles, energy in the form of food was required. The suitcase could have been lifted by an elevator. Energy in the form of electricity (in turn deriving from oil, coal, gas, solar and other renewable sources) would then be required for the useful work needed to be done, that is, to lift the suitcase and the elevator cage.
The law of conservation of energy
The law of conservation of energy states that energy is neither created nor destroyed – it changes from one form to another. Most texts show this by the example of conversion of potential to kinetic energy.
Potential energy of a body is dictated purely by its position, relative to a base. Thus if a body of mass M is situated at height h above the surface of the Earth, then its potential energy (P.E.) with respect to the surface of the Earth is M*g*h. or PE = M*g*h.
This potential energy converts into kinetic energy (or the energy of a body arising from its motion), when it is dropped from the height h. Let us see what happens when it is dropped.
All bodies on Earth are continually subjected to the force of gravity. Structures that support, in some fashion, keep them from falling to the surface. As they fall, they are continually accelerated, if we do not take into account the resistance from the air. Thus a stone drops faster than a piece of paper, as the paper encounters more air resistance. For this example, let us not consider air resistance.
The speed V with which the body of mass M will hit the ground when dropped from the height h may be found from the formula:
V*V = u*u + 2*g*S, and putting u=0 (as the mass M was not moving initially) and S = h, we have V*V = 2*g*h.
Thus h = V*V/(2*g).
Substituting this value of h in the equation for potential energy, we have the term M*g*V*V/(2*g) or M*V*V/2. This term we call Kinetic Energy, or KE of a body. Thus the potential energy of the mass (which is now lost) is converted into another form of energy, which is kinetic energy, or the energy it now has due to its motion. (Note that this velocity is with respect to the surface of the earth, and that is the inertial reference frame. Further note that there is only this one inertial reference frame – none other.)
Entropy and the Law of Conservation of Mass and Energy
It is not with Potential Energy, but with Kinetic Energy, that Work is really done. So if body M fell upon one end of a see-saw, it could lift another mass up, and so exert force upon it and do measurable work. However, if it just fell onto the ground, it would do no work at all. All the PE and later KE would just become heat, via impact. As an engine, it would work at zero efficiency. The inability to capture the energy which dissipated as heat, and to do work with it, is understood in terms of the thermodynamic term “entropy”. This is a mathematical concept, and roughly quantifies the extent of disorder in the cyclical system involved in a particular engine’s way of working. Thus an engine which converts a higher percentage of energy into work, creates less entropy than one which does not. Also, entropy helps us to understand the law of conservation of energy in these terms – the potential and kinetic energy that existed were not “destroyed”; these two forms of energy were converted into heat energy. that increased the entropy of the universe. And ultimately, with usage, all kinetic energies become such useless heat energy, and thus the entropy of the universe keeps on increasing. Consequently, the universe is always “running down” from order to disorder – this is proven by every thermodynamic process.
The law of conservation of energy got a jolt with the discovery of the radio-active process. It could not explain why a piece of matter could continuously give out energy, in the form of radiation. To explain this, Einstein’s famous equation, e=mcc, was used to form a new law, that of the law of conservation of mass and energy. The energy from radiation was explained: it had to result from a loss of mass. Also, the energy from the Sun, till then a puzzle, was explained in terms of loss of solar mass, as a result of its presumed internal nuclear reactions. We will deal with this concepts later in this treatise.
Energy is stored in the form of fossil fuel, water and other matter at high altitudes, under the Earth’s surface as geo-thermal energy, and so on. It also pours endlessly from the Sun, and solar energy is the main reason for most activity on Earth. According to current thinking in physics, all such energy has always existed, though it can and does change its form (from chemical to kinetic to potential to heat and so on). It can also be formed from the extinction of matter (theoretically or apparently, from nuclear reactions). Finally, energy can never be created from nothing. Such is modern, conventional wisdom, expressed as the second law of thermodynamics.
The Creation of Energy with Internal Force
Let us assume that a body can be moved with internal force, the way our flying saucer works as described earlier. Then we can show, that if this sort of motion is possible, then we do create energy from nothing other than the capacity of the body to accelerate itself with internal force. And thus violate the second law of thermodynamics, which roughly speaking, is a restatement of the law of conservation of energy.
We have seen, that a body of mass M accelerating with internal force can reach the speed of N*V where
N is the number of internal hits, and V is the velocity. Then after reaching the velocity N*V, with respect to the original inertial frame of reference, the kinetic energy of the body is 0.5*M*(N*V)*(N*V) or 0.5*M*N^2*V^2.
Now let us take the position that the energy required to raise the velocity of the body from 0 to V, and from V to 2*V, from 2*V to 3*V is always the same, with the application of internal force. For, we are going from a rest inertial frame of reference, to a V velocity inertial frame of reference, to a 2*V inertial frame of reference, and so on. The same energy E is always spent when we go from one inertial frame of reference, to another with the same level of velocity separation. To understand this, is the tricky bit underlying our entire perspective, and we will explore this in detail, in the subsequent paragraphs.
The Travelator Analogy
To take an analogy, let there be N heavy travelators (flat moving platforms, of the type commonly found in modern airports for faster passenger movements) running side by side, with speeds v, 2v, 3v and so on with respect to the Earth – but just for the purpose of this example we assume that none of these are actually connected to the Earth. (We will consider the case when they are actually being supported by the Earth, later.) They are from our assumption effectively like long spaceships, N in number, and flying side by side, parallel, with speeds, v, 2v, 3v, ,,, Nv.
Let a person of mass M jump from rest to the travelator running a speed v. The travelator will accelerate him to the speed v in time t. When it will reach that speed v, the kinetic energy of the body with respect to the earlier inertial frame of reference (the rest frame, or 0*v frame) is 0.5*m*v*v.
Then after he has attained balance, he jumps to the next travelator running at 2*v. Does the travelator running at 2*v spend exactly E = 0.5*m*v*v amount of energy, with respect to the earlier 1*v frame, in raising the speed of the body from v to 2v, or does it spend rather more as it now has to push the mass over a longer distance to make it reach the 2v speed? This is the most crucial question, and there are two very different answers that we get, and they are very important.
To reach the speed 2v from v, a greater distance S2 has to be covered than the distance to reach the speed v from initial velocity u=0, call that S1. The latter distance is from the equation S = u*t + 0.5*a*t*t is S1 = 0.5*a*t*t. While the former is S2 = v*t + 0.5*a*t*t. As v = a*t, S2 = 1.5*a*t*t. When jumping from 2v to 3v inertial frame of reference, the initial velocity is 2*a*t, so S3 will be equal to 2a*t*t + 0.5a*t*t = 2.5*a*t*t and so on, till SN will be (N-0.5)*a*t*t.
From the earlier discussion on the definition of Work, on the face of it, it will appear that with the application of the same Force F acting over the increasing distances S1, S2, S3… an increasing amount of work needs to be done, in raising the velocities from v to 2v, 2v to 3v, and so on. And since the energy required for doing so has to be at least equal to the work that is done, we must need increasing amounts of energy in such raising of velocities.
The above scenario, is actually valid when the travelators are fixed on the ground. For in that case, it is exactly what happens when potential energy is converted into kinetic energy – a constant acceleration (meaning constant force) acts over the entire distance, and thus the kinetic energy gained at the end (0.5*M*N^2*V^2) will be equal to the work done upon it by this force. So there is no net gain of energy when the travelators are fixed to the earth, as it is reaction from the earth which is always acting as the external force, pushing the mass to higher and higher velocities over longer and longer space intervals.
But when the travelators are not fixed to the earth, that is to say, the earth (nor the air nor water, as for airplanes and ships) can not provide a reaction, the situation changes drastically. When the person steps on the first travelator from the inertial rest frame, he gains the acceleration from the force exerted upon him by the travelator. As the travelator is now not fixed to the earth, this means that its own velocity reduces very slightly – depending upon its mass with respect to the person who boards it. This drop in speed (v_small) can be found from the law of conservation of momentum. Here MT denotes the mass of the travelator.
So, MT*v + M*0 = (MT + M)*(v – v_small)
or MT*v = MT*v – MT*v_small + M*v – M*v_small
or v_small = M*v/(MT + M)
This happens with the velocities taken with respect to the rest or 0*v inertial frame of reference.
A similar phenomenon happens when the person moves from the 1*v reference to the 2*v reference – the velocity of the 2*v reference decreases as much as the 1*v reference had decreased. We assume here that before jumping from 1*v to 2*v, the original speed v of the travelator had been regained, through acceleration of the travelator with internal force from its own internal energy (like say, an electric battery). Later, we will investigate how this may be done, in practice.
MT*2v + M*v = (MT + M)*(2v – v_small), or
v_small = M*v/(MT + M)
Similarly, equal internal energy (say like an equal fixed loss in power to each battery, or equal amount of fuel utilized) is being spent by the all the other travelators, in accelerating the person upto the v, 2v,… velocity level from v – v_small, 2v – v_small,…, as the velocity differential v_small is always the same. Now why is this so? (I repeat, here at this stage we assume that the travelator is capable of acceleration with just its own internal energy.)
Consider the mathematics involved. Look at the equations for the terms S1, S2, S3… SN. They are of the form S = v*t + 0.5*a*t*t, where v is of the form a*t, 2a*t, 3a*t, and so on. Now, with external force, rooted in the the rest v=0 inertial frame of reference, these are the components of the S terms relating to the distance over which the external force has to be applied. But with internal force accelerating the mass, over successive inertial frames of reference, these terms automatically vanish, and become zero. The only effective distance over which the internal force actually works is always the constant second term of S, namely 0.5*a*t*t, for every single travelator. However, to the external observer on the rest v=0 inertial frame of reference, it will appear that this internal force is being applied over the entire S distance, and so, a lot of work is being done for which a lot of energy must be required. But this is just an appearance. When a child in an airplane travelling at 1000 Km per hour with respect to the ground throws a ball at 10 Km per hour within the airplane in the direction of travel, then the ball travels at 1010 Km per hour with respect to the ground. Now, no one on the ground - not even the greatest Olympian - can have the energy to throw the ball at 1010 Km per hour! But a child can have the energy to throw the ball at 1010 Km per hour with respect to the ground, when the child is situated in an inertial frame of reference (the airplane) that is travelling at 1000 Km per hour with respect to the ground.
Simply because the travelators are not fixed to the earth, we find that on the whole a great deal less energy is being expended to accelerate a body from the velocity 0 to the velocity N*v. In short, more work is being done on a body, than the energy being spent upon it, in the scenario where there is transition of the body between successive independent inertial reference systems existing in parallel at speeds v, 2v, 3v…Nv.
In each such transition of the body, there is an increase in velocity of the body, by the amount v. This happens over the distance 0.5*a*t*t. The force F = M*a acts over this distance, so the work done on the body is F*distance = M*a*0.5*a*t*t, or 0.5*M*v*v. The energy required for doing this much work is at least that much, or a factor greater than 1, call it k, multiplied by that much. So the internal energy expended for each transition is E = 0.5*k*M*v*v. And the sum of internal energies expended for the whole system is N*E or 0.5*k*N*M*v*v. Now the total kinetic energy of the body M after N such transitions, and with respect to the rest (v=0) frame of reference, is 0.5*M*N*N*v*v. There is a difference between these two values, and that is the extra energy that has been created with this process.
The True Mass-Energy Relationship
This difference, call it extra or free energy, or energy created EC, is (0.5*M*N*N*v*v - 0.5*k*N*M*v*v), or 0.5*M*N*v*v(N – k). Thus
Energy created (EC) = 0.5*M*N*(N – k)*v^2.
For a large value of N, and low k, this could be a large value. We will later use this equation to show how we get energy from any explosion, whether it be the striking of a match, or a bomb, or a nuclear explosion. We now see exactly how energy is created, when there is material interaction between co-existing inertial frames of reference at different relative velocities. We will use this formula (along with the ways we derived it) later to show how the heavenly bodies (the sun and the stars, and our earth and the planets, and even our moon) create energy, directly and continuously and endlessly.
Energising our Flying Saucer (SpaceShip) to Light Speed
It was not easy to directly present the physics for the energies related to the motion of a flying saucer, to my intelligent 13 year old reader, so in the earlier section an indirect and more understandable approach was taken, with the N travelators running side by side with continuously ascending relative velocities.
Consider now all the travelators merged together as a spacehip of mass M. This spaceship has the capacity to accelerate from 0 to v, v to 2v, 3v to 4v, and up to Nv, with its internal force, and using its internal energy. We have seen how each transition takes the energy E; N transitions will take the energy N*E.
Let us see how much energy is required to take a spaceship of total mass 100,000 Kgs to light speed.
We take the earlier values of, v = 10 m/s, N = 3*10^7, and let k = 2.
Then using the formula N*E or N*0.5*M*v*v*k, we have
0.5 * 100,000 * 3*10^7*10*10*2 MKS units of energy required to make the flying saucer reach light speed.
Since the time required to expend this energy will be 3*10^7 seconds (or nearly one year), the power required in watts will be 0.5*100,000*10*10*2 = 10^7 or 10 MegaWatts. A small nuclear power generator can easily do that!
Thus we see, interstellar travel is perfectly possible in theory – and in practice too, if and only if acceleration of a body with internal force is possible. If and only if our New Physics, that assumes that a body can be moved by internal force without pushing back anything, or pushing out its own mass, is proved to be correct via rigorous experiment, that is. We will now investigate this most crucial aspect.
The Movement of Bodies with Internal Force
Newton’s First Law as it now exists – every body stays at rest or in uniform motion in a straight line, unless acted upon by an external force – was developed long before the study of electromagnetism took place.
The author took a keen interest in finding out if a body could be moved by internal force, by purely mechanical means. He made a machine (using wood, wheels, threaded rods and bolts, and bricks) in which he himself could sit, with another person, and do pushing actions (thus expending internal energy) and taking the frictional effects with the ground into consideration, but without any drive to the wheels. Ultimately it was found that while rotating and counter-rotating motions are possible, with such actions, within the machine, it was not possible to change the centre of gravity of the system.
So mechanical action could not be expected to move a body with internal force – Galileo and Newton had done a thorough job! But what about electro-mechanical action? The laws of motion had been codified long before electromagnetism was discovered.
About the discovery of electromagnetism (or the relationship between electricity and magnetism), now. Magnetism was known for ages – the naturally occurring magnet lodestone was used as a compass by sailors. Electrostatics (the attraction – or clinging forces - between different sorts of matter, when they were rubbed against each other, such as paper and glass) was also known, and Leyden jars were used for special effects. However the concept of the electric current was unknown.
It may seem strange to think, but it was a dead frog which led to practically all the great advantages and conveniences we enjoy today – right from computers and iPhones and robots to electric light and motors and microwave ovens! In 1786, Professor Luigi Galvani of the University of Bologna, in an anatomical experiment upon the nerves of a dead frog found that the frog’s muscles contracted when its nerves where touched by scissors in an electric storm. Professor Alessandro Volta, an outstanding professor of physics at the University of Pavia, coined this term as “galvanism” and described Galvani’s work in these words: “it contains one of the most beautiful and most surprising discoveries”. What Prof. Galvani had actually done was to cause an electric current to flow, and to note its effects. His work stimulated Prof. Volta to discover a source of constant current electricity through principles derived from physics and chemistry. It was the voltaic pile, or electric battery. All subsequent usages of electric power stem from this invention.
The Electric Current
It was soon found out that when a current existed in a conductor (like, a copper wire) which was placed in a magnetic field (or the area over which the magnet exerts its measurable or significant attractive effects), that conductor experienced a force. The direction of the current, the direction of the magnetic field, and the direction of the force – were all at right angles to each other. Also, when a conductor was moved by a force in a magnetic field, a potential to create electric current (or voltage) was generated by the conductor, along its two ends (terminals). Thus, we have the electric motor, and the electric generator, respectively.
Whenever a current existed in a conductor, a magnetic field was immediately created around it. This field could be increased in strength if the conductor was looped, to form a coil. More the number of coils, and more the current, the stronger the magnetic field. For a given voltage, across the terminals of the conductor, if the resistance of the conductor is very low, we can generate very high currents, as there is a relation between the voltage V, current I, and resistance R, namely V = I*R.
Linear Motion, Momentum, Force, Energy, Internal Force Engines, and the design of Interstellar Spacecraft
Inertial Frame of Reference
We must first consider an “inertial frame of reference”. For most practical terrestrial purposes any point notionally fixed on the Earth is valid as an inertial frame of reference. Speaking more generally, an inertial frame of reference is a specified point, line, area or volume such that all points that are fixed on that particular frame of reference do not move with respect to each other; and such an inertial frame of reference is not accelerating away from another such inertial frame of reference. Two points fixed on the Earth do not move with respect to each other (unless there is an earthquake or continental drift in which case a re-mapping is required). So the fixed points on the Earth constitute a frame of reference, roughly referred to as “ground”.
What velocity a particular frame of reference have with other such frames of reference may or may not be known. Thus we can never know the velocity of the Earth, at any point on its surface, with respect to the fixed, static, all-pervading and solid medium which has been defined as ether; through which all electromagnetic waves have to travel. (For while we may find out the orbital speed around our Sun, we cannot know the speed of the solar system in the galaxy, the speed of the galaxy itself, or the speed of the intergalactic cluster our galaxy belongs to, as they all progress in space.) On the other hand, we can know the velocity of an airplane or clouds with respect to the ground (the inertial reference frame for the aircraft), with aid of such a tool as radar. But for the passengers and crew in the airplane, who are far away from the ground, the inertial reference frame relevant to them is their moving airplane itself. Thus, no matter what the speed of the airplane with respect to its inertial reference frame – the ground directly below it at any time – the distance between the passenger and say the door of the airplane will remain constant. And if the passenger is still, his speed with respect to the airplane (his inertial reference system) will be zero, while with respect to the ground directly below him, his speed will be the same as that of the airplane.
One-dimensional space
The following equations that will be derived, are essentially in one-dimensional space. Multi-dimensionality, involving vector algebra, is essentially for practical engineering issues – the consideration of multi-dimensional space will make the understanding of the fundamental equations of motion more complex, without adding any further necessary illumination at this preliminary stage. After all, to go to the stars, all we have to do is to point the nose of our starship to the required star, and just go there! That would be a very uni-dimensional path, indeed! So for the purpose of understanding, so far as this present article is concerned, we limit our inertial frame of reference not over the whole curved planet, but the small part of it around us that is flat or linear (as is the case for most of our day to day life, unless we happen to be astronauts in far outer space, in which case our initial frame of reference is the point that is the centre of the Earth). In any case, making the mathematics simple should make this article understandable to anyone of acute intelligence above 13 years of age, and with some knowledge of algebra.
Fundamental Equations of Motion
The fundamental equations of motion, upon which most engineering are ultimately based are very simple. I will show how simple they are, by deriving them from the first principles, below.
1. s = v_avg*t. s is the distance covered over the time t by a body moving with average velocity v_avg (average rate of change of distance with time) along the distance. Thus if a body is moving with an average speed of 10 meters per second in a straight line, in 5 seconds it moves 50 meters away from the fixed point it was at 5 seconds ago.
2. v = u + a*t. v is the final velocity of the body in motion, u is the initial velocity, t is the time period over which the body changes from velocity u to velocity v without change in direction, and a is the average acceleration (rate of change of velocity with time) over the time period along the straight line. Thus if a body moves with a speed of 10 meters per second past the reference point, accelerates continuously at 2 meters per second per second on the average (like a car, say), after that, then after 10 seconds of undergoing such acceleration it will have the velocity of 10+2*10 = 30 meters per second.
These two equations written above are the most basic. It should be clear that they do not deal with any turning motion – merely motion in a straight line.
The following two equations are derived from the above.
3. s = u*t + 0.5*a*t*t (the variables have been defined earlier, but here the acceleration value is constant throughout the time period t.)
Derivation:
v_avg = (u+v)/2, if the acceleration is uniform over the time t.
s = v_avg*t = ((u+v)/2)*t = ((2*u+ at)/2)*t = u*t + 0.5*a*t*t.
If u=0, or the body starts from rest, then s = 0.5*at*t.
4. v*v = u*u + 2*a*s (the variables are as defined earlier, and here too a is constant over the time interval between u and v).
Derivation:
v*v = (u + at)*(u + at) = u*u + 2uat + a*a*t*t = u*u + at(2u +at)
= u*u +at(u + v) = u*u + at*2*v_avg = u*u + 2*a*t*v_avg
= u*u + 2*a*s
The above four equations are the most fundamental to describe linear motion in what is known as the Newtonian frame of reference. They have been know for centuries.
I would like to point out their extreme simplicity and elegance to the reader.
Momentum and Rocket Science
We now come to the very important concept of momentum of a body. By body we mean an amount of matter contained within a closed geometry. A closed geometry is like, say, a box or a ball.
Now, what is matter? Matter is anything that can exert force, when linearly accelerated, upon other matter; or have its smallest components vibrate, as a consequence of travelling electromagnetic waves. It requires force for acceleration. (We will talk about force, later.) All matter is influenced by the force of gravity, which is always present on, near and within the Earth. From our Earthly perspective, matter is whatever on Earth that has weight.
The quantity of matter contained within the body, is called its mass. Units of mass are grams, ounces, tons, kilograms, etc.
The momentum of a body is simply the product of its mass and velocity. Thus M = m * v, where M is the momentum, m is the mass and v is the velocity of the mass. Just as the velocity is relative to an inertial reference system, the momentum is also thus relative to the inertial reference system.
The above equation is simple. What is slightly more difficult to understand is the concept of the law of conservation of momentum. The definition of the law of conservation of momentum is simple, though. It says that the momentum of a system of bodies in motion is always constant. So M = m(1)*v(1) + m(2)*v(2) +…. +m(n)*v(n), where the m(1), v(1) terms refer to the mass and velocity of the first body, and so on. In our simple one-dimension world, the v() terms could have only positive or negative signs, meaning the masses going one way, or the opposite way.
Fundamentals of Rocket Science (in terms of purely momentum) expressed in (relatively) simple mathematics
It is from this simple law of conservation of momentum, that the fundamentals rocket science may be discovered. Suppose the question is, what is the size of the rocket which will take a piece of mass (in outer space, away from large bodies such as the Sun or the Earth) to the speed of light, and bring it back to its rest speed? The law of conservation of momentum will let us find that out!
Note: Please leave out this section for now, if you feel somewhat challenged by the mathematics!
Now by the law of conservation of linear momentum, when a body of mass m is ejected from a body of resting mass M (that is, its velocity is zero with respect to the inertial reference frame) in some given inertial reference frame, then the mass M gets a velocity V of magnitude m*v/(M-m) and in the direction opposite to v, where v is the velocity of the ejected mass m. For M*0=0=mv – (M-m)V, or V/v=m/(M-m).
Using this basic approach, let us have:
a large rocket with a payload pointing at a star to which it must go;
the rocket is at rest, in the inertial reference frame, call it the first inertial reference frame;
an engine which ejects bodies (gases, whatever) at a constant velocity v in the opposite direction of travel of the rocket;
the bodies ejected decrease in mass at a proportionate rate, say at r percent of the existing rocket mass with every ejection.
Then we have for the first ejection: m = M*r, and so m*v = M*r*v,
and M*r*v= (1-r)M*V from the law of conservation of momentum, or V=r*v/(1-r); for ejected mass m = M*r, and (1-r)*M is the balance mass, for M*(1-r) + r*M = M.
Also, if v/V = X, or the number of times the speed of the ejecting mass is greater than that of the increased rocket velocity, then X = (1-r)/r. Then rX = 1-r, or r = 1/(1+X).
After this the rocket of mass M(1-r) will move at velocity V, indefinitely, if there is no impediment. The new inertial frame of reference is this rocket of mass M*(1-r) moving at speed V.
Let it now further throw out a mass r(1-r)*M out at speed v, or momentum r(1-r)*M*v, with respect to this new inertial frame of reference.
Then the mass of the rocket left will be M*(1-r) - r(1-r)*M, or M*(1-r)(1-r).
So the velocity gained by the rocket will be r(1-r)**v/(M*(1-r)(1-r)) = rv/(1-r) = V.
Then, with respect to the rocket's earlier speed, that is V, the speed of the rocket will increase by r(1-r)*M*v / M*(1-r)(1-r) or r*v/(1-r) which is the same amount as earlier, that is V. With respect to the first inertial reference frame, is speed now will be 2*V.
After two ejections, the mass of the rocket is (1-r)^2*M; we can see this way that after N ejections it will be (1-r)^N*M. And the speed of the rocket will be N*V, with respect to the first inertial reference frame.
Let us now try to calculate the size of the rocket that will take us to the speed of light (which is what we must reach if we are ever to go to the nearby stars).
Now c/V, (c is the speed of light) is the value of N, that is, the number of mass ejections required for the payload to reach light speed.
Let the mass of the rocket, which was M at zero speed with respect to the star it wanted to reach, at light speed be N. Then from the above derivation N=(1-r)^(c/v) * M.
And M, the mass of the rocket at zero speed, will be:
M = N/((1-r)^(c/V)).
Putting the value of r = 1/(1+X), the final formula for the mass M of the rocket required to send a payload N to the speed of light is:
M = N/(1-(1/(1+X)))^(c/V), where X = v/V
Now c = 3*10^8 m/sec. If V = 30 m/sec, then c/V = 10^7. If v = 3000 m/sec, then X = 100, and (1-r) = 0.99. And (c/V) factor is 10^7.
Putting these values we get M = N/(.99)^10000000, an extremely high number, which even the powerful modern calculator cannot display. To get some idea, let us use old fashioned log tables. The log of .99 is -0.0043648, so the log of (.99)^10000000 = -43648. Taking antilog of -43648, we find that the factor M/N is 10^43648, which is 1 followed by 43648 zeroes! Now, the integer Google, which is 1 followed by a 100 zeroes, is a sufficiently large mind boggling number – this one baffles belief! Can we make our rocket smaller?
Only if X is of the order of c/V, is there any chance of making a rocket of reasonable size. For this, let us see
When X=c/V, then (1-r) = (1 – 10^(-7)) or 0.9999999, Then M = N/(0.9999999)^10000000 = N/0.3678 = 2.71828 N.
So the mass expelled for the rocket to reach light speed is M – N = 2.71828N – N = 1.71828 N
Well, this will take us to light speed. It can coast at this speed for a while, say a few years. But then it has to slow down. This can be done be turning around 180 degrees, and decelerating by throwing out mass the other way to what was being done previously. So to return to rest speed (with respect to the first inertial reference frame) we have N = 2.71828 * Q, where Q is the final payload at rest in the faraway star.
So the size of the rocket to deliver the payload Q will be M = 2.71828*N or 2.71828^2 * Q = 7.389 Q. So as much as 6.389 times Q (final payload) will have to be spent as rocket fuel mass to be ejected. Not too efficient, what? Besides, what with the trail of matter following the rocket’s linear path, later trips could well be highly jeopardized or well-nigh impossible, what with great risk of collision.
Still, this does not look too bad! Is there a catch anywhere? Unfortunately, there is one.
When X = 10^7, and V=30m/sec, then the speed of the exhaust mass v =V*X has to be 3*10^8 m/second, which is the speed of light! To throw out mass at light speed, is beyond the scope of modern science and technology. But only if we throw out mass at that speed, can we reach light speed for the rocket. But how to accelerate any mass to light speed in the first place!
Thus we find, mathematically, that it is impossible to go to the nearest star, or reach light and faster than light speed, with rocket technology. It is simply out of the question! Either we make a rocket of impossibly large size, or we throw out mass at light speed. So, are we really stuck for ever on Planet Earth? Let us see! Having understood the concept of momentum, we shall now try to understand the concept of force, in the sense of physics.
Force
The basic definition of force is given by Leonardo da Vinci, one of the greatest geniuses of all time, in one of this notebooks.
“Force I define as a spiritual power, incorporeal and invisible, which with brief life is produced in those bodies which as the result of accidental violence are brought out of their natural state and condition. I have said spiritual because in this force there is an active, incorporeal life; and I call it invisible because the body in which it is created does not increase in weight or in size; and of brief duration because it desires perpetually to subdue its cause, and when this is subdued it kills itself.”
Thus Leonardo defines force in metaphysical terms. Force emanates from the realm of the spirit. It is the constant interaction between the Divine and the material – and its key characteristic is life. The word Leonardo used for force, “spiritual power”, may for clarity better be replaced as “spiritual manifestation”; as “power” has a meaning different from “force” in the science of dynamics. (Specifically, power is rate of usage of energy, and energy as we will see can be measured in terms of force multiplied by distance.)
Force is essentially something mysterious. It is the ultimate show of life whenever life forms are involved, or otherwise there is something Divine, Purely Random or Divinely Sanctioned Random quality about force, when non-living objects are involved. Just as mass is what is moved by force, force is what moves mass.
On earth, we have to continuously experience the force of gravity – one of the most fundamental forces of Nature. This means, that if a body is let go near the surface of the earth, it will accelerate constantly towards the surface of the earth with an acceleration value of g or roughly 9.8 meters per second per second.
However, Newtonian mechanics is not based upon the necessity of the Earth. It is abstract – when applied to Earth it takes in the gravitational pull of the Earth. So when we talk of Newton’s laws of motion, we do not have to take the Earth into account as a must. They apply everywhere.
The concept of mass comes into Newton’s laws. Mass is simply a lump of matter, which could exist as solid, liquid, gas or in a charged state known as plasma. Matter is what is moved, or can be moved, with force. A particular piece of solid matter is taken as a certain standard of mass. Such standards known are the kilogram, pound, gram, ton, etc.
We ignore Newton’s first and third laws, for now, and concentrate upon the second law. The second law defines force mathematically in terms of the concepts of momentum.
The second law of Newton is stated as: Rate of change of momentum is equal to the applied force and takes place opposite to the direction in which the force is applied. So if the mass of the body is M, and the initial velocity is u, then its initial momentum is M*u. If now a force acts upon it, causing acceleration, then after a time t over which this force is acting, the velocity of this body is v. And its momentum is thus m*v. The change of momentum is thus m*u – m*v, or m*(u-v). The rate of change of momentum is m*(u-v) divided by the time over which this change took place, which is t. So the rate of change of momentum is m*(u-v)/t. And by the second law, this is equal to the applied force F, which has been acting for t seconds upon the mass.
So the equation is F = -1*m*(u-v)*t, the minus sign coming because the force is opposite to the direction of rate of change of momentum, thus
F = m*(v-u)/t = m*a. Thus the force is always in the direction of the acceleration.
This is a crucial equation. To see its use, let us see (below) how this equation alone can explain the movement of rockets.
Force in Rocketry
As we know, a rocket works when mass is ejected out of it at high speed. It goes in the opposite direction to which mass is ejected. When it ignited, hot gas comes out, but it is only after a while that the rocket starts to rise from the Earth. That happens when the force exerted by the exiting gases are equal to the (at that time) weight of the rocket. As the gases leave the rocket, the mass of the rocket also decreases by that amount. If the gases are leaving the rocket at a constant rate of v meters per second with respect to the rocket’s end, then the momentum they exert upwards will be m(t)*(v(t)-0) or m(t)*v(t), where the inertial reference frame is the rocket itself. And force F at time t, that is, the rate of change of momentum, will be F(t) = time derivative (d/dt) of m(t)*v(t) that is
F(t) = m(t)*d(v(t))/dt + v(t)*d(m(t))/dt.
Some elementary knowledge of differential calculus is necessary to understand how the above equation was derived. I request my clever 13 year old reader, who knows algebra but has yet to study calculus, to simply accept this, for now!
m(t) or the mass coming out of the rocket at time t over a very small time interval dt approaches zero as dt (the very small interval of time) approaches zero, so the first term in the above equation tends to zero in the limit. Also, if v(t) is a constant, then the term d(v(t)/dt has to be zero. On the other hand, the term d(m(t))/dt has a finite value – it is equal to the loss of mass of the rocket over a particular time period. Thus the equation reduces to
F(t) = v(t)*d(m(t))/dt , or the velocity of the escaping gases with respect to the rocket multiplied by the rate of loss of mass of the rocket.
To give some values, if the exiting speed of gases is 1000 m per second, and mass is lost at 1 Kg per second, then the force exerted upwards will be 1000 MKS units (newtons). That will support a mass of little more than 100 Kgs (thus give it “antigravity”). If say the speed of the exiting gases is increased to 1200 m per second, with the same rate of loss of mass, then a net upward force of 200 newtons will be exerted upon the mass of 100 Kgs – thus giving it a 2 meter per second per second acceleration. So, the mass will go up with increasing speed.
All this should square with our personal experiences with rockets. In NASA launches, we see gases come out for a while before the rocket starts to rise. To begin with, the gases do not come out with such speeds as would be necessary to provide the lift. As they do so, the rocket ascends faster and faster. But, it has to keep on losing mass to do so. We have seen earlier, there is no way it can ever reach light speed, and so, rocket science can never take us much beyond the solar system.
Must we be tied to Earth and the Solar System at most?
Modern physics cannot have it otherwise! We have found out that it is impossible to construct a rocket to reach light speed. Apart from this, there are certain theories relating to relativity that make it impossible to reach the speed of light. Apparently, strange things happen when we are nearing light speed – mass becomes extremely massive, length becomes near zero… We will talk about these concepts later.
If we think a bit more deeply, we will find that it is actually our pre-occupation with rocket science that is preventing us from reaching the nearby stars. Cannot there be any other, or more advanced science, that can overcome the issues that prevent us from reaching the stars? This was dealt with in detail in articles and a book published in Internet by the author nearly ten years ago – the name of the book is “To the Stars!” and is freely available (since around 2000 AD) in the website www.users.bigpond.com/adda1234/index.htm The original concepts first laid out there, form the basis of what may be popularly known as anti-gravity; and these basic principles show the way to go to the stars. We will here deal with them in brief.
Internal Force (Anti-Gravity)
So how does a man in a rocket actually get off the ground? The answer is, as given earlier, when the force exerted upward by the exhaust gases are more than the rocket’s weight. The gases are created by using up the energy within the rocket.
Suppose there is a different way to create upward force, using up energy stored or generated within the rocket, of which we do not yet know anything about. We further assume that this upward force did not involve loss of matter from the rocket.
So Upward Force U > M*g, where M is the mass of our spaceship (yes, it is not a rocket ship any more) and g is the acceleration due to the gravity of our planet. Putting U = M*a, then M*a > M*g, and so the net force N that is accelerating the spaceship up is M*(a-g). As the spaceship goes further away from earth, the value of g decreases till it becomes nearly zero. When it does so, we are then accelerating with the acceleration a, now up from the initial value (a-g).
Reaching light speed
Equation 2 in Section 1 is
V = u + a*t
When v is the speed of light, and u is zero (with respect to the initial reference frame) and a is say slightly more than g or 10 m/sec/sec, then
3*10^8 = 10*t, or it takes 3*10^7 seconds to reach light speed. Which comes out to 347 days, or slightly less than one year.
Reaching the nearest star
The nearest star is a little over four light years away. So after reaching light speed, or exceeding it, the spaceship can coast for say 3 years. Then it turns around 180 degrees till it faces the Earth, and starts the engines as it did. This time this will be deceleration, and it will take another 347 days for the spaceship to come to rest at the destination star. So overall, about 5 years will be required to make a one-way trip to the nearest star!
Newton’s Third Law
Thus, if only we do not think in terms of throwing out mass at high speed, in order to accelerate a vehicle, then only can we reach light speed.
Is it possible to accelerate a vehicle without throwing out mass? We do have cars, and bicycles. These do not accelerate by throwing out mass, nor are animals needed to pull them. How do these work? These use internal force, from the internal energy from the cyclist’s muscles or the fuel needed for the internal combustion engine.
Cars and bicycles do use internal force, but they cannot operate without external force. The movement of the wheels upon the ground causes frictional forces to operate – the ground is pushed back by the wheels, and the ground pushes the car or bicycle forward. The more strongly the ground is pushed backed, with equal vigor the car or bicycle is pushed forward by the ground. In short, without the whole Earth (or at least a substantial part of it) to push back, there is no chance of forward motion for the car or bicycle. So if we raise the rear wheel of the bicycle using a stand, and pedal fast and furiously, we go nowhere – only the wheel spins faster and faster!
About airplanes, now, that do use internal force and do not use friction. The propeller in the airplanes push back the air, and with as much force that the air is pushed back, the air pushes the airplane forward. Similarly a jet engine sucks in air, heats it by burning fuel in it, and then pushes out the heated and polluted air with great force – usually greater than the propeller airplanes, and so jet engines cause airplanes to fly faster. No airplane can fly where there is no air (such as in outer space, where you can accelerate only if you push out part of your own mass), just as cars and bicycles cannot move if they cannot push back the earth with their wheels.
From the above we find examples of the truth that every force exerts an equal and opposite force, for all known mechanical phenomenon involving force. This has been found to be universal and so far (according to all the scientific books and papers on the physics of motion, save just one, about which we will talk about later) has no exceptions. It is a law of nature – according to all the natural philosophers.
Newton’s First Law of Motion
We now have made clear the fundamental concepts of momentum and force. There is another very important fundamental concept, inertia. Isaac Newton dealt with this concept very carefully. Inertia is defined as the property of the material body to stay still, or to move with a constant velocity, in the absence of any forces affecting the body. Every material body, it has been found, does normally exist in a state of inertia. So universal is this phenomenon, that it is deliberated upon as a law of nature, and is stated as “Every body stays in a state of rest, or in a state of uniform motion in a straight line, unless affected by an externally impressed force.”
So a car moves when the earth pushes it (this is the external force) when there is frictional contact between the wheel and the ground; the airplane moves when the air forced back by the propeller motion pushes it; the rocket moves when the gases pushed out from it, push back the rocket.
Changing Newton’s First Law
The word “externally” in the definition of the First Law of Motion has somehow managed to insinuate itself there, detracting thus from the concept of inertia. Suppose we simply delete it, and see what are the consequences. Then the First Law of Motion will be: “Every body stays in a state of rest, or in a state of uniform motion in a straight line, unless affected by an impressed force.”
This implies, that some energetic activity within the body, can give it motion without any need for friction, for pushing out mass, or pushing back mass. In other words, it behaves like a flying saucer is expected to behave – it just takes off!
Motion of a Flying Saucer, or Future Space-Ship
Let us try to get a little mathematical once again, in order to better understand the flight of the flying saucer, or space-ship. Suppose a net force F acts on the body, this force arising from the energies arising from within the body, for a time t. The mass of the ship is M. Then the acceleration a will be F/M. Since this acceleration acts over the time period t, after the time t, the velocity of the ship will be from the formula v = a *t,
V = (F/M) * t.
So after reaching the speed V from the initial zero velocity (with respect to the inertial frame of reference), if all internal activity ceases, the ship will continue to move with the speed V with respect to that original inertial frame of reference. Now this is something like firing a bullet in deep space: from zero velocity with respect to the gun, originally, the bullet keeps on going at speed V with respect to the gun, for all time to come (unless it hits a meteor or comet or planet or star).
Now after reaching the speed V, the activity same as what was done before again takes place within the ship – a Force F acts for a time t on the ship. Now the inertial frame of reference for this force, is the space-ship moving with constant speed V (this speed V is with respect to the original inertial frame of reference). Thus this force will cause a further addition of velocity, V, to the ship. With respect to the original inertial frame of reference, its speed will now be V+V=2V, after two “hits”.
Thus after N such hits, the speed of the flying saucer will be N*V. If V is 10 meters/ second, then we have seen earlier how light speed can be reached in less than one year, at this rate.
Travel to the Moon
However to go to the moon or other planets, we do not need to fly that fast, at the speed of light. Let us see how long it will take us to go to the moon, with this ship, using the laws of motion see earlier. The distance from the Earth to the Moon is 400,000 Km.
To travel half the distance or 200,000 Km, or 200,000,000 meters accelerating at 10m/sec will be from the formula S = u*t + 0.5*a*t*t, putting u=0,
200,000,000 = 0.5*10*t*t, or t*t = 40,000,000, or t = 6324seconds.
To travel the remaining 200,000 Km, by turning 180 degrees and decelerating at the same rate, will take the same time. So the total time for travel to the moon will be 12648 seconds, or around four hours. The maximum speed reached will be from the equation v = u + a*t, or v = 0 + 10*6324 = 63,240 meters per second.
The concepts of Energy, Engine and Work
Crucial to the design of flying saucers powered with Internal Force Engines, is our understanding of the concepts of energy, engine and work.
We had in the previous section sneaked in the concept of “Energy” from which the Force is generated. What is this energy?
Energy is another very fundamental concept, like mass, momentum, and force. It is the source of much talk in the world today. It’s usage (whether to get it from fossil fuels or renewable sources such as the solar, wind, tidal, geo-thermal, hydroelectric, etc.) is much debated. However few people other than practical engineers have a clear idea about the nature of energy, in the scientific sense.
Essentially Energy is defined in terms of the Work that is done by a body. As much Work is done, at least so much energy is required. I say “at least” because usually only a fraction of the energy is converted into useful work by any engine – the remaining energy is usually lost in the form of “heat”; or rather, in the dissipation of energy as heat, flowing into the universe, as it cannot be converted into work, or converted into other forms of energy.
I did mention above the word “engine” – so what is an engine? An engine is a mechanical or electro-mechanical device which converts energy (from whatever source) into work. A heat engine is one which converts heat energy into work. The earliest example of the heat engine is the steam engine. Water was heated by burning wood or coal; thus high-pressure steam was generated, and this steam was used to push a piston up and down a cylinder; and this linear movement was converted into rotary movement by mechanical contraption to rotate wheels. Now a lot of heat that was generated was not converted into work – thus the heat engine had low efficiency. An engine that uses electrical energy to produce work (an electrical motor that is) is a lot more efficient than a heat engine.
So what is Work? This is the key question. Work is measured by the following formula: Work = Force * Distance, where the Distance (an unit of length) must be aligned in the direction of the force. In other words, if a force F moves a body of mass M over a distance D, then the Work Done is F*D, or M*a*D, where F = M*a.
What work is done in carrying say a suitcase and also the the carrying person totaling mass M along a flat corridor? The person who carried it surely must think he has done some work, for his muscles tell him something! But according to our formula, no work has been done! For the force that has been acting on the suitcase, gravity, is at right angles to the direction of the motion of the suitcase. But if the person carries the suitcase up a flight of stairs, of height say D, then he has done the Work measured in terms of M*g*D, where g is the acceleration due to the Earth’s gravity, and the force acting upon the suitcase.
Back to the concept of energy, now. The energy to lift the suitcase (and the person too) came from the carrier’s muscles. To sustain the muscles, energy in the form of food was required. The suitcase could have been lifted by an elevator. Energy in the form of electricity (in turn deriving from oil, coal, gas, solar and other renewable sources) would then be required for the useful work needed to be done, that is, to lift the suitcase and the elevator cage.
The law of conservation of energy
The law of conservation of energy states that energy is neither created nor destroyed – it changes from one form to another. Most texts show this by the example of conversion of potential to kinetic energy.
Potential energy of a body is dictated purely by its position, relative to a base. Thus if a body of mass M is situated at height h above the surface of the Earth, then its potential energy (P.E.) with respect to the surface of the Earth is M*g*h. or PE = M*g*h.
This potential energy converts into kinetic energy (or the energy of a body arising from its motion), when it is dropped from the height h. Let us see what happens when it is dropped.
All bodies on Earth are continually subjected to the force of gravity. Structures that support, in some fashion, keep them from falling to the surface. As they fall, they are continually accelerated, if we do not take into account the resistance from the air. Thus a stone drops faster than a piece of paper, as the paper encounters more air resistance. For this example, let us not consider air resistance.
The speed V with which the body of mass M will hit the ground when dropped from the height h may be found from the formula:
V*V = u*u + 2*g*S, and putting u=0 (as the mass M was not moving initially) and S = h, we have V*V = 2*g*h.
Thus h = V*V/(2*g).
Substituting this value of h in the equation for potential energy, we have the term M*g*V*V/(2*g) or M*V*V/2. This term we call Kinetic Energy, or KE of a body. Thus the potential energy of the mass (which is now lost) is converted into another form of energy, which is kinetic energy, or the energy it now has due to its motion. (Note that this velocity is with respect to the surface of the earth, and that is the inertial reference frame. Further note that there is only this one inertial reference frame – none other.)
Entropy and the Law of Conservation of Mass and Energy
It is not with Potential Energy, but with Kinetic Energy, that Work is really done. So if body M fell upon one end of a see-saw, it could lift another mass up, and so exert force upon it and do measurable work. However, if it just fell onto the ground, it would do no work at all. All the PE and later KE would just become heat, via impact. As an engine, it would work at zero efficiency. The inability to capture the energy which dissipated as heat, and to do work with it, is understood in terms of the thermodynamic term “entropy”. This is a mathematical concept, and roughly quantifies the extent of disorder in the cyclical system involved in a particular engine’s way of working. Thus an engine which converts a higher percentage of energy into work, creates less entropy than one which does not. Also, entropy helps us to understand the law of conservation of energy in these terms – the potential and kinetic energy that existed were not “destroyed”; these two forms of energy were converted into heat energy. that increased the entropy of the universe. And ultimately, with usage, all kinetic energies become such useless heat energy, and thus the entropy of the universe keeps on increasing. Consequently, the universe is always “running down” from order to disorder – this is proven by every thermodynamic process.
The law of conservation of energy got a jolt with the discovery of the radio-active process. It could not explain why a piece of matter could continuously give out energy, in the form of radiation. To explain this, Einstein’s famous equation, e=mcc, was used to form a new law, that of the law of conservation of mass and energy. The energy from radiation was explained: it had to result from a loss of mass. Also, the energy from the Sun, till then a puzzle, was explained in terms of loss of solar mass, as a result of its presumed internal nuclear reactions. We will deal with this concepts later in this treatise.
Energy is stored in the form of fossil fuel, water and other matter at high altitudes, under the Earth’s surface as geo-thermal energy, and so on. It also pours endlessly from the Sun, and solar energy is the main reason for most activity on Earth. According to current thinking in physics, all such energy has always existed, though it can and does change its form (from chemical to kinetic to potential to heat and so on). It can also be formed from the extinction of matter (theoretically or apparently, from nuclear reactions). Finally, energy can never be created from nothing. Such is modern, conventional wisdom, expressed as the second law of thermodynamics.
The Creation of Energy with Internal Force
Let us assume that a body can be moved with internal force, the way our flying saucer works as described earlier. Then we can show, that if this sort of motion is possible, then we do create energy from nothing other than the capacity of the body to accelerate itself with internal force. And thus violate the second law of thermodynamics, which roughly speaking, is a restatement of the law of conservation of energy.
We have seen, that a body of mass M accelerating with internal force can reach the speed of N*V where
N is the number of internal hits, and V is the velocity. Then after reaching the velocity N*V, with respect to the original inertial frame of reference, the kinetic energy of the body is 0.5*M*(N*V)*(N*V) or 0.5*M*N^2*V^2.
Now let us take the position that the energy required to raise the velocity of the body from 0 to V, and from V to 2*V, from 2*V to 3*V is always the same, with the application of internal force. For, we are going from a rest inertial frame of reference, to a V velocity inertial frame of reference, to a 2*V inertial frame of reference, and so on. The same energy E is always spent when we go from one inertial frame of reference, to another with the same level of velocity separation. To understand this, is the tricky bit underlying our entire perspective, and we will explore this in detail, in the subsequent paragraphs.
The Travelator Analogy
To take an analogy, let there be N heavy travelators (flat moving platforms, of the type commonly found in modern airports for faster passenger movements) running side by side, with speeds v, 2v, 3v and so on with respect to the Earth – but just for the purpose of this example we assume that none of these are actually connected to the Earth. (We will consider the case when they are actually being supported by the Earth, later.) They are from our assumption effectively like long spaceships, N in number, and flying side by side, parallel, with speeds, v, 2v, 3v, ,,, Nv.
Let a person of mass M jump from rest to the travelator running a speed v. The travelator will accelerate him to the speed v in time t. When it will reach that speed v, the kinetic energy of the body with respect to the earlier inertial frame of reference (the rest frame, or 0*v frame) is 0.5*m*v*v.
Then after he has attained balance, he jumps to the next travelator running at 2*v. Does the travelator running at 2*v spend exactly E = 0.5*m*v*v amount of energy, with respect to the earlier 1*v frame, in raising the speed of the body from v to 2v, or does it spend rather more as it now has to push the mass over a longer distance to make it reach the 2v speed? This is the most crucial question, and there are two very different answers that we get, and they are very important.
To reach the speed 2v from v, a greater distance S2 has to be covered than the distance to reach the speed v from initial velocity u=0, call that S1. The latter distance is from the equation S = u*t + 0.5*a*t*t is S1 = 0.5*a*t*t. While the former is S2 = v*t + 0.5*a*t*t. As v = a*t, S2 = 1.5*a*t*t. When jumping from 2v to 3v inertial frame of reference, the initial velocity is 2*a*t, so S3 will be equal to 2a*t*t + 0.5a*t*t = 2.5*a*t*t and so on, till SN will be (N-0.5)*a*t*t.
From the earlier discussion on the definition of Work, on the face of it, it will appear that with the application of the same Force F acting over the increasing distances S1, S2, S3… an increasing amount of work needs to be done, in raising the velocities from v to 2v, 2v to 3v, and so on. And since the energy required for doing so has to be at least equal to the work that is done, we must need increasing amounts of energy in such raising of velocities.
The above scenario, is actually valid when the travelators are fixed on the ground. For in that case, it is exactly what happens when potential energy is converted into kinetic energy – a constant acceleration (meaning constant force) acts over the entire distance, and thus the kinetic energy gained at the end (0.5*M*N^2*V^2) will be equal to the work done upon it by this force. So there is no net gain of energy when the travelators are fixed to the earth, as it is reaction from the earth which is always acting as the external force, pushing the mass to higher and higher velocities over longer and longer space intervals.
But when the travelators are not fixed to the earth, that is to say, the earth (nor the air nor water, as for airplanes and ships) can not provide a reaction, the situation changes drastically. When the person steps on the first travelator from the inertial rest frame, he gains the acceleration from the force exerted upon him by the travelator. As the travelator is now not fixed to the earth, this means that its own velocity reduces very slightly – depending upon its mass with respect to the person who boards it. This drop in speed (v_small) can be found from the law of conservation of momentum. Here MT denotes the mass of the travelator.
So, MT*v + M*0 = (MT + M)*(v – v_small)
or MT*v = MT*v – MT*v_small + M*v – M*v_small
or v_small = M*v/(MT + M)
This happens with the velocities taken with respect to the rest or 0*v inertial frame of reference.
A similar phenomenon happens when the person moves from the 1*v reference to the 2*v reference – the velocity of the 2*v reference decreases as much as the 1*v reference had decreased. We assume here that before jumping from 1*v to 2*v, the original speed v of the travelator had been regained, through acceleration of the travelator with internal force from its own internal energy (like say, an electric battery). Later, we will investigate how this may be done, in practice.
MT*2v + M*v = (MT + M)*(2v – v_small), or
v_small = M*v/(MT + M)
Similarly, equal internal energy (say like an equal fixed loss in power to each battery, or equal amount of fuel utilized) is being spent by the all the other travelators, in accelerating the person upto the v, 2v,… velocity level from v – v_small, 2v – v_small,…, as the velocity differential v_small is always the same. Now why is this so? (I repeat, here at this stage we assume that the travelator is capable of acceleration with just its own internal energy.)
Consider the mathematics involved. Look at the equations for the terms S1, S2, S3… SN. They are of the form S = v*t + 0.5*a*t*t, where v is of the form a*t, 2a*t, 3a*t, and so on. Now, with external force, rooted in the the rest v=0 inertial frame of reference, these are the components of the S terms relating to the distance over which the external force has to be applied. But with internal force accelerating the mass, over successive inertial frames of reference, these terms automatically vanish, and become zero. The only effective distance over which the internal force actually works is always the constant second term of S, namely 0.5*a*t*t, for every single travelator. However, to the external observer on the rest v=0 inertial frame of reference, it will appear that this internal force is being applied over the entire S distance, and so, a lot of work is being done for which a lot of energy must be required. But this is just an appearance. When a child in an airplane travelling at 1000 Km per hour with respect to the ground throws a ball at 10 Km per hour within the airplane in the direction of travel, then the ball travels at 1010 Km per hour with respect to the ground. Now, no one on the ground - not even the greatest Olympian - can have the energy to throw the ball at 1010 Km per hour! But a child can have the energy to throw the ball at 1010 Km per hour with respect to the ground, when the child is situated in an inertial frame of reference (the airplane) that is travelling at 1000 Km per hour with respect to the ground.
Simply because the travelators are not fixed to the earth, we find that on the whole a great deal less energy is being expended to accelerate a body from the velocity 0 to the velocity N*v. In short, more work is being done on a body, than the energy being spent upon it, in the scenario where there is transition of the body between successive independent inertial reference systems existing in parallel at speeds v, 2v, 3v…Nv.
In each such transition of the body, there is an increase in velocity of the body, by the amount v. This happens over the distance 0.5*a*t*t. The force F = M*a acts over this distance, so the work done on the body is F*distance = M*a*0.5*a*t*t, or 0.5*M*v*v. The energy required for doing this much work is at least that much, or a factor greater than 1, call it k, multiplied by that much. So the internal energy expended for each transition is E = 0.5*k*M*v*v. And the sum of internal energies expended for the whole system is N*E or 0.5*k*N*M*v*v. Now the total kinetic energy of the body M after N such transitions, and with respect to the rest (v=0) frame of reference, is 0.5*M*N*N*v*v. There is a difference between these two values, and that is the extra energy that has been created with this process.
The True Mass-Energy Relationship
This difference, call it extra or free energy, or energy created EC, is (0.5*M*N*N*v*v - 0.5*k*N*M*v*v), or 0.5*M*N*v*v(N – k). Thus
Energy created (EC) = 0.5*M*N*(N – k)*v^2.
For a large value of N, and low k, this could be a large value. We will later use this equation to show how we get energy from any explosion, whether it be the striking of a match, or a bomb, or a nuclear explosion. We now see exactly how energy is created, when there is material interaction between co-existing inertial frames of reference at different relative velocities. We will use this formula (along with the ways we derived it) later to show how the heavenly bodies (the sun and the stars, and our earth and the planets, and even our moon) create energy, directly and continuously and endlessly.
Energising our Flying Saucer (SpaceShip) to Light Speed
It was not easy to directly present the physics for the energies related to the motion of a flying saucer, to my intelligent 13 year old reader, so in the earlier section an indirect and more understandable approach was taken, with the N travelators running side by side with continuously ascending relative velocities.
Consider now all the travelators merged together as a spacehip of mass M. This spaceship has the capacity to accelerate from 0 to v, v to 2v, 3v to 4v, and up to Nv, with its internal force, and using its internal energy. We have seen how each transition takes the energy E; N transitions will take the energy N*E.
Let us see how much energy is required to take a spaceship of total mass 100,000 Kgs to light speed.
We take the earlier values of, v = 10 m/s, N = 3*10^7, and let k = 2.
Then using the formula N*E or N*0.5*M*v*v*k, we have
0.5 * 100,000 * 3*10^7*10*10*2 MKS units of energy required to make the flying saucer reach light speed.
Since the time required to expend this energy will be 3*10^7 seconds (or nearly one year), the power required in watts will be 0.5*100,000*10*10*2 = 10^7 or 10 MegaWatts. A small nuclear power generator can easily do that!
Thus we see, interstellar travel is perfectly possible in theory – and in practice too, if and only if acceleration of a body with internal force is possible. If and only if our New Physics, that assumes that a body can be moved by internal force without pushing back anything, or pushing out its own mass, is proved to be correct via rigorous experiment, that is. We will now investigate this most crucial aspect.
The Movement of Bodies with Internal Force
Newton’s First Law as it now exists – every body stays at rest or in uniform motion in a straight line, unless acted upon by an external force – was developed long before the study of electromagnetism took place.
The author took a keen interest in finding out if a body could be moved by internal force, by purely mechanical means. He made a machine (using wood, wheels, threaded rods and bolts, and bricks) in which he himself could sit, with another person, and do pushing actions (thus expending internal energy) and taking the frictional effects with the ground into consideration, but without any drive to the wheels. Ultimately it was found that while rotating and counter-rotating motions are possible, with such actions, within the machine, it was not possible to change the centre of gravity of the system.
So mechanical action could not be expected to move a body with internal force – Galileo and Newton had done a thorough job! But what about electro-mechanical action? The laws of motion had been codified long before electromagnetism was discovered.
About the discovery of electromagnetism (or the relationship between electricity and magnetism), now. Magnetism was known for ages – the naturally occurring magnet lodestone was used as a compass by sailors. Electrostatics (the attraction – or clinging forces - between different sorts of matter, when they were rubbed against each other, such as paper and glass) was also known, and Leyden jars were used for special effects. However the concept of the electric current was unknown.
It may seem strange to think, but it was a dead frog which led to practically all the great advantages and conveniences we enjoy today – right from computers and iPhones and robots to electric light and motors and microwave ovens! In 1786, Professor Luigi Galvani of the University of Bologna, in an anatomical experiment upon the nerves of a dead frog found that the frog’s muscles contracted when its nerves where touched by scissors in an electric storm. Professor Alessandro Volta, an outstanding professor of physics at the University of Pavia, coined this term as “galvanism” and described Galvani’s work in these words: “it contains one of the most beautiful and most surprising discoveries”. What Prof. Galvani had actually done was to cause an electric current to flow, and to note its effects. His work stimulated Prof. Volta to discover a source of constant current electricity through principles derived from physics and chemistry. It was the voltaic pile, or electric battery. All subsequent usages of electric power stem from this invention.
The Electric Current
It was soon found out that when a current existed in a conductor (like, a copper wire) which was placed in a magnetic field (or the area over which the magnet exerts its measurable or significant attractive effects), that conductor experienced a force. The direction of the current, the direction of the magnetic field, and the direction of the force – were all at right angles to each other. Also, when a conductor was moved by a force in a magnetic field, a potential to create electric current (or voltage) was generated by the conductor, along its two ends (terminals). Thus, we have the electric motor, and the electric generator, respectively.
Whenever a current existed in a conductor, a magnetic field was immediately created around it. This field could be increased in strength if the conductor was looped, to form a coil. More the number of coils, and more the current, the stronger the magnetic field. For a given voltage, across the terminals of the conductor, if the resistance of the conductor is very low, we can generate very high currents, as there is a relation between the voltage V, current I, and resistance R, namely V = I*R.
pnal...@gmail.com
May 27, 2019, 12:57:05 AM5/27/19
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Still bat-shit crazy, I see...
Arindam Banerjee
May 29, 2019, 10:30:26 PM5/29/19
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On Monday, 27 May 2019 13:23:15 UTC+10, Arindam Banerjee wrote:
> Section 1
>
> Linear Motion, Momentum, Force, Energy, Internal Force Engines, and the design of Interstellar Spacecraft
>
> Inertial Frame of Reference
>
(section 1 will be continued in another post)
> Section 1
>
> Linear Motion, Momentum, Force, Energy, Internal Force Engines, and the design of Interstellar Spacecraft
>
> Inertial Frame of Reference
>
Cheers,
Arindam Banerjee
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