To Androcles
St.M
story of yours.
M.
Androcles
"St.M" <st....@gmail.com> wrote in message
news:1126275552.5...@z14g2000cwz.googlegroups.com...
I kept a copy. You can google for the thread.
Sam, Joe, a mosquito and a ladder.
by Androcles
Much of this story is credited to Daryl McCullough, only the ladder
and the story was added by me. It explains the origins of Einstein's
Special Relativity scaled for those having difficulty grasping the
subject.
Scale 1 ft : 60,000 km
Sam and Joe are housepainters, and are walking along the street at 3 fps
in still air carrying a 32 ft long ladder between them, Joe leading the
way. Sam is carrying some paint cans and Joe has the brushes and
rollers.
At some point along their journey a mosquito named Albert buzzes past
Sam's ear. Sam swats at it, but drops a can of red paint as he does so.
Albert the mosquito flies along the ladder from Sam to Joe at a constant
speed of 5 fps. When it reaches Joe, Joe also swats at it, but drops a
paint roller. Albert, still hungry but not liking the smell of Joe's
cigar,
flies back along the ladder toward Sam, again with a constant speed of
5 fps in the still air. Upon reaching Sam, once again Sam tries to swat
the wee beastie but drops a can of green paint. He yells as the mosquito
bites him and this startles Joe, who drops a paint brush.
Now it's your turn. I'll give the answers further down, but take a
moment to do the calculations for yourself.
1) How many seconds did it take for Albert to fly from Sam to Joe?
2) How many seconds did it take for Albert to fly from Joe to Sam?
3) How far is it between the red paint can and the roller?
4) How far is it between the green paint can and the roller?
(Answers below)
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Assume the speed of the mosquito is c = 5 fps.
The speed of Sam and Joe is v = 3 fps, given.
We then must have a distance along the road for Joe of
32ft + vt, and for the mosquito, a distance of ct.
Solving for t,
ct = 32 + vt
ct - vt = 32
t(c-v) = 32
t = 32 /(c-v) = 32/(5 - 3) = 16 seconds
So the answer to Q.1) is 16 seconds.
The mosquito coming back is going to meet Sam going forward,
so it flies along the 32 feet of the ladder in time
t = 32/(c+v) = 32/8 = 4 seconds.
The answer to Q.2) is therefore 4 seconds.
The distance from the dropped red paint can to the dropped roller
is just ct, or 5 * 16 = 80 feet, so the answer to Q.3) is 80 ft.
Or we could do it by vt + 32 = 3 * 16 + 32 = 80, once again.
(Remember Joe had a 32 ft head start over the mosquito)
Coming back, Albert again flies at 5 fps but this time
for only 4 seconds, so it reaches the green paint can 20 feet
from the roller, which is the answer to Q.4)
So, as Sam sees it, Albert takes 16 seconds to reach Joe, flying at
5-3 = 2 fps, and 4 seconds to return, flying along the ladder at
5+3 = 8 fps.
Now we think like Einstein with his mosquito brain. Sam wants to know
when the mosquito reached Joe.
He isn't able to see the mosquito, its too small at 32 feet away,
so he guesses that since it went 32 ft each way, and took 20 seconds to
fly away and back again, it must have reached Joe after 10 seconds = ½
of 20.
So we explain it carefully. First we label the red paint can "A" and the
dropped roller "B". We write:
If at the point A of space there is a clock, an observer called Sam at
the red paint can will determine the time values of events in the
immediate proximity of the red paint can by finding the positions of
the hands which are simultaneous with these events. If there is at the
point B of space another clock in all respects resembling the one at
the red paint can, it is possible for an observer Joe at the dropped
roller
to determine the time values of events in the immediate neighbourhood
of the roller at B. But it is not possible without further assumption to
compare, in respect of time, an event at A with an event at the dropped
roller, B. We have so far defined only an "A time" and a "B time."
We have not defined a common "time" for the red paint can and the
dropped roller, for the latter cannot be defined at all unless we
establish
by *definition* that the "time" required by a mosquito to travel from
the red paint can to the dropped roller equals the "time" it requires to
travel from B to the red paint can, A.
Note the *definition*. Remember this is hypothetical, not real. The
definition is very important.
Now, we want to do this algebraically, because tomorrow Joe and Sam
might be carrying a different length of ladder, running at a different
speed, whatever, and we want a general solution.
So we write:
If we place x'=x-vt, it is clear that a point at rest in the system
ladder must have a system of values x', y, z, independent of time.
What that means is the ladder's length is x', so that
32 = 80 - 3 * 16,
and doesn't change as time passes. Did you think it would? Well,
we'll have to see. Maybe if we water it, it might grow.
According to Albert, we are to assume the speed of the mosquito is
independent of the speed of Sam (which is fair enough) and also we are
to assume that the time for the mosquito to make the round trip (20
seconds) when divided by 2 is equal to the time it took to reach Joe,
16 seconds, by Albert's DEFINITION.
We don't know yet about the 16 seconds, we can only write it
algebraically and pretend it is 10 seconds.
It is actually written as x'/(c-v) [or 32/(5-3) in real numbers].
Now we say:
From the origin of system ladder (Sam's end) let a mosquito be emitted
at the time tau0 along the ladder to x' (Joe's end), and at the time
tau1
be reflected thence (that just means go back) to the origin of the
co-ordinates (which we are deliberately vague about as to whether
we mean Sam on the ladder or the red paint can), arriving there at the
time tau2; we then must have (don't you just love that phrase, "then
must have" ?) ½(tau0 + tau2) = tau1,
or ½([midmorning + 0] + [midmorning + 20]) = [midmorning + 16],
which is curious to say the least, since Sam and Joe could be doing this
in the late afternoon for all the difference it would make.
But ok, Einstein wanted to be complete, so I guess its fine.
But our hero and physics wizard isn't satisfied with this. Oh no, we
need to include the length of the ladder as well, or we won't have any
spacetime to prattle on about later so that people will see just how
smart we are.
It is very important to include the length of the ladder into the
equation. You'll see why later.
Here is Einstein's equation:
½[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v))
You can read about it at
http://www.fourmilab.ch/etexts/einstein/specrel/www/
(in Section 3)
Putting in the mosquito numbers,
½[tau(0,0,0,t)+tau(0,0,0,t+32/(5-3)+32/(5+3))] = tau(32,0,0,t+32/(5-3))
½[tau(0,0,0,t)+tau(0,0,0,t+20)] = tau(32,0,0,t+16)
In agreement with experience (gotta love that phrase!) clearly!
(0,0,0,t) is pretty meaningless, and we can drop the "t+" since we
really don't care if Sam and Joe are walking in the midmorning or
late afternoon.
So, (by the vector addition of (0,0,0,0) + (0,0,0,20) !)
½ * tau(0,0,0,20) = tau(32,0,0,16).
Now do you see why we need to include the length of the ladder into the
evaluation of time? We can't just say ½ * 20 = 16 without it. Even my
grandson would say that wasn't right, and he's not learning algebra yet.
There's some differentiation by Einstein to make himself look smart and
important, he has to show off all his skills if not his common sense,
because "common sense is the collection of prejudices acquired by age
eighteen", or so he tells us, and he eventually arrives at
tau = (t-vx/c^2) / sqrt( 1 - v^2/c^2 )
xi = (x-vt) / sqrt( 1 - v^2/c^2 )
eta = y
zeta = z.
This is is the procedure so far:
1)
Define t = x'/(c-v) = x'/(c+v)
because the time for light to go from A to B equals the time it takes
to travel from B to A
2)
½[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v))
3)
½[1/(c-v)+1/(c+v)] * dtau/dt = dtau/dx' + 1/(c-v) * dtau/dt
4)
dtau/dx' + v/(c^2-v^2) * dtau/dt = 0
5)
tau = a * ( t - (vx' / (c^2-v^2)))
6)
tau = (t-vx/c^2) / sqr(1-v^2/c^2)
That is what you get when you treat time as if it were a vector and mix
in some distance. Time is NOT a vector quantity, it has no inverse.
There is no identity, no minus time to when you were.
We can forget y and z, the mosquito didn't fly up into a tree or into
the ditch at the side.
We apply this to the equations derived:
tau = (16 - 3 * 80 / 25) / sqrt (1 - 3^2/5^2)
= (6.4) / 0.8
= 8 seconds
xi = 32 / sqrt (1 - 9 / 25)
= 40 feet
Sanity check:
c = 40 ft / 8 seconds = 5 fps. Yep, that's the right speed for Albert.
So...
We are standing at the roadside watching Sam and Joe carry a 32 ft
ladder that they think is a 40 ft ladder, because the speed of
mosquitoes is 5 fps in all inertial frames of reference, and
½ * 20 * 0.8 = 8 seconds without batting an eyelid.
There is a slight hitch, though. In the roadside frame, where is A, at
the red paint can or the green paint can? We still want half the time
to travel from A to B to be equal to the time to travel the round trip,
and the distance from the red paint can to the roller is 80 ft, and back
to the green paint can is 20 ft.
Which is the "origin of the coordinates"?
This is best satisfied by beefing up Albert's speed to infinity, and as
Einstein says, the velocity of light... err... mosquitoes in out theory
plays the part, physically, of an infinitely great velocity."
It must be right, its only algebra after all is said and done.
So now you should be able to fully understand Special Relativity, all
you need do is replace the speed of the mosquito with the speed of
light, have Sam and Joe run at the relativistic speed of 0.6c, the
algebra is perfect, and who needs common sense anyway?
Just remember that 32 ft ladders stretch to 40 ft ladders when you run
with them at 180,000 km/sec, and you'll be as smart as Einstein the
cretin.
For myself, I'll keep the collection of prejudices I acquired by the
time I was eighteen, since that defines common sense.
Androcles
St.M
Dirk Van de moortel
"St.M" <st....@gmail.com> wrote in message news:1126500069.7...@f14g2000cwb.googlegroups.com...
> Thanks a lot...
Preparing a Master's thesis on Usenet Village Idiots?
Dirk Vdm
St.M
;-)
Dirk Van de moortel
"St.M" <st....@gmail.com> wrote in message news:1127209669.2...@g44g2000cwa.googlegroups.com...
> ;-)
Don't forget to send me a copy then :-)
Dirk Vdm
Henri Wilson
As the main character, you'll deserve a signed copy.
>
>Dirk Vdm
>
>
HW.
www.users.bigpond.com/hewn/index.htm
see: www.users.bigpond.com/hewn/variablestars.exe
"Sometimes I feel like a complete failure.
The most useful thing I have ever done is prove Einstein wrong".