Maximum Acceleration Unblocked

[Ping Weafer]

Ihave attempted to read a lot of books and articles titled "The Physics of Tennis." I like reading physics and math books in bed because they put me to sleep almost immediately. Please understand that I have always loved physics. I wanted to be a physicist until I discovered during my freshman year in college in 1972 that as a result of the hippies convincing the country to dismantle NASA and the military-industrial complex that physicists were mowing lawns to live. That, and there probably isn't enough caffeine in the world to get me through earning a Ph.D. in physics. I tell you this because I do not intend to bore you with any esoteric discussions of how the ball and strings deform at the moment of contact resulting in non-elastic blah, blah, blah. That doesn't help you to enjoy the game. Mostly this page is designed to support some of the outlandish claims made elsewhere in the site. I am not even going to support my assertions with references. If you doubt any of the concepts I refer to you can look them up yourself. Sleep tight!

Physics is the study of things that move. Things that move have kinetic energy. Their energy can be used to do work, like moving electrons in a motor that is lifting an elevator against gravity. A gallon of gasoline can also be used to do work, but it doesn't move. It has potential energy.

Things that have mass and move also have momentum which is the ability to change how other things move. Momentum has direction - it is a creature of space. Energy has no direction - it is a creature of time.

To change an object's energy or momentum you must apply a force (push, pull, attract, repel etc.) No force - no change, everything would remain the same forever. You can't create momentum. You can only steal it from something else. When we hit a tennis ball, we must steal momentum from the earth and use it to change the ball's direction. (The earth changes its motion in a direction opposite that of the ball.) That means we have to push off on the earth before we hit the ball.

Forces acting on an object in opposite directions tend to cancel each other out. What is left over is a net force which can then accelerate the object. Hold a bowling ball in your hand, and you are pushing up an amount equal to the gravity pulling it down. Increase the force, and the ball accelerates up, decrease it, and the ball falls.

To change the direction of a ball we need to apply a force for a length of time. The longer the time or greater the force, the greater the change in the direction of the ball. The change of the balls flight path is in the direction of the applied force. A force applied for duration of time is called impulse.

When a moving racket meets a moving ball with no force applied there is a change in the direction of the ball, so there must be an impulse applied to the ball. This impulse comes from an exchange of momentum between the racket head and the ball. Because the racket head has more mass than the ball, the racket loses some velocity but doesn't change its speed or its direction, very much (velocity is speed in a given direction). The ball is light and therefore changes its velocity quite a bit, actually reversing one of its directions - the direction that is directly opposed by the racket. Its other velocity component, the one at right angles to the racket direction, is unchanged. Thus a ball that was coming in and down before impact will be traveling out and down after impact - perhaps directly into the base of the net. A non-accelerating racket meeting a moving ball is an example of an 'elastic collision.' An elastic collision occurs when you just swing at the ball or stick your racket out and let the ball hit it. Arranging an elastic collision between racket and ball gives you very little control over the direction of the ball after the strike. To control the ball you must inject impulse aimed only in the direction you want the ball to go.

The ultimate direction of the ball after an elastic collision is way beyond our control since it depends as much on things we do not control - such as the incoming direction and spin on the ball - as it depends on the stuff we do control such as the speed, direction and angle of the racket face. Therefore to control a ball one must introduce a force through the racket at the moment of contact with the ball. That moment lasts only about .05 seconds. The applied force must pass through the racket but must not cause the racket to move so much that you mis-hit the ball. Given time, any force you apply to the racket to add spin and control will misdirect the racket sweet spot away from the ball. The forces that carry control and extra spin must, therefore, be released all at once, over .05 seconds, starting just before the moment of contact - like an explosion. To accomplish this, you pull the racket face to the ball during the lag phase, then let go and allow the racket to coast into the ball as the forces stored in the forearm explode into the ball. Ideally, the forces that add control should be stored in your arm somehow and released suddenly just before the moment of contact. That way they do not perturb the actual path of the racket head until after the ball has already left the strings, i.e., in the follow through, so they can't interfere with the all-important mission of addressing the ball.

This is an essential concept to accept if you are ever going to "get" elite tennis stroke technique. When you throw the tennis racket back in the backswing portion of any stroke, then start to drag it forward it gets heavy. Really heavy.

The racket's recalcitrance is just fine since the inertia of the racket is what winds up the forearm to store the muscular force that will be used to inject spin and directional control into the ball at the moment of contact. But when I say it gets really heavy, I mean really, really heavy.

The average racket weighs about 500 grams (g) or .5 kilograms (kg). An elite player laying into a forehand can accelerate that racket from 0 to 35 meters per second (m/s) ( = 78 miles per hour) in .25 seconds (s). The acceleration of the racket is therefore 35 m/s .25 s = 140 m/s/s . OK, so what? The easiest way to visualize this is to consider that the earth's gravitational acceleration g = 10 m/s/s. Gravitational acceleration is what gives things "weight." That means that since the racket is accelerating at 14 times the force of gravity, it feels like it weighs 14 times as much! That would be 7 kg or about 15 pounds! About half of this weight is in the head of the racket, and since the head is some distance from your wrist, it creates a considerable moment of inertia. Moment of inertia reflects the reality that accelerating a rock on the opposite end of a stick requires more force the longer the stick. While the moment of inertia of the racket head can present challenges, it also provides the opportunity to create leverage with mechanical advantage.

Mechanical advantage is a measure of force amplification by a lever. It measures the force that has to be applied to the one end of a stick (or lever arm) to accelerate a weight attached at the other end. A mechanical advantage of 1.0 means the same force appears at both ends; a ten-pound force can move a ten-pound weight. A mechanical advantage of 1.0 also means that moving either end of the lever one foot moves the other end one foot. A mechanical Advantage of 2.0 means twice the force but half the distance of translation at the far end (like a crowbar) and .5 means half the power but twice the translation at the far end (like a sword). In a tennis stroke, we are always dealing with a mechanical advantage of less than one. The farther the racket head is from your body, the lower the mechanical advantage.

Based on the discussion of racket inertia above, you know that the racket puts up quite a fight early in the acceleration phase, so to deliver the maximum amount of acceleration early on it makes sense to have the lever arm as short as possible, i.e. with a bent elbow and the racket lagging behind the wrist. This is a very comfy position for accelerating a racket, and it is sometimes hard to get non-athletes (like me) to give up that position later in the stroke when a longer lever arm is needed. The result is often a cramped stroke with limited power and pace on the ball and lots of shanks off of the top of the racket. We all have the ability to adjust the distance of the racket head from our bodies by simply extending the elbow and wrist as the ball is accelerating. Increasing the length of the lever arm reduces mechanical advantage. Why is this necessary?

Once acceleration is underway the racket goes back to feeling and handling like a 10-ounce racket, and at that point, all we have to do is guide the sweet spot around into the ball, but we want more. We want ultimate power. Power production by our muscles depends on muscle velocity - the speed at which the muscle is contracting. When our limbs move faster, the muscles cannot keep up, so they generate less power and less force. That is why some bicycles have 15 or more gears, and even on the flats, one tends to upshift as the bike goes faster. Higher gears have a lower mechanical advantage meaning the feet can go slower while the wheels can go faster. Ideally, we want to maximize power from the legs regardless of the speed of the bike. In tennis, we reduce mechanical advantage by gradually increasing the distance of the racket head from our bodies as we accelerate the racket. Lengthening the lever arm thereby reducing mechanical advantage as the stroke progresses maximizes the transfer of power from the muscles to the ball.

Again, the key to applying leverage in tennis is learning to adjust the moment of inertia of the racket head by changing the geometry of the hitting arm throughout the stroke. Starting with a bent elbow and fully lagged wrist, we first extend the elbow then straighten the wrist as the racket head accelerates. Once the racket head is moving, our goal is to give it that little extra boost to get it going even faster, and that requires an extended arm and an un-lagged wrist - a longer lever.

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