I have 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!

A (Very Brief) Intro to Physics

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.

Things that aren't moving don't want to be moved. Once moving, they want to keep on moving in the same direction. We call that inertia.

Impulse

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.

It is this mysterious and magical stored force that injects control force into the ball and is therefore the ultimate and only reliable source of control.

Racket Head Inertia

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.

So heavy in fact that your forearm muscles are unable to prevent the racket head from lagging behind the wrist. Let's see how heavy we are talking about, shall we?

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

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?

Applying Leverage

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.

Spin

In addition to moving from place to place, objects can rotate in space. Rotation follows many of the same rules as linear motion. There is rotational inertia, rotational velocity, rotational momentum, and rotational energy. There is also rotational impulse which is one way we can change an object's rotational direction and speed (rotational velocity). Rotational impulse occurs whenever a force is applied to an object at some distance from its center of mass (think of turning a crank). That force is called torque. Whenever you strike a ball with a racket face that is moving in one direction but pointing in another, you apply torque to the ball and introduce spin. We call this brushing across the ball.

Spin is useful in tennis; it changes the flight path of the ball, affects the bounce and makes the ball come off of your opponent's racket at unexpected angles. The problems with brushing across the ball are twofold. First, you cannot apply pure torque in this way. The friction against the ball tends to change the ball's trajectory in the direction that you are brushing. If you try to increase your topspin by pulling up faster or at a greater angle, you tend to direct the ball up and over the baseline. More importantly, adding severe spin through brushing makes it harder to make solid contact with the ball since instead of a direct head-on collision it is a glancing blow. The solution to this dilemma is to add some spin with brushing and the rest using rotational impulse.

Rotational Impulse

Rotational impulse adds spin by applying a torque force at the moment of contact analogous to adding linear impulse. The racket does not have to be rotating at all to inject spin; it just has to be trying to rotate. The trick is to store rotational force in the arm during the backswing, carry it intact through the swing, and then release it just as the racket meets the ball.

The Aerodynamics of Spin

A moving ball that is spinning will follow a curved path through the air due to a phenomenon called the Magnus effect. The rotation changes the direction of air rushing over the ball. Topspin directs it upward, and slice directs it downward. Since every action has an equal and opposite reaction, underspin tends to lift balls in opposition to gravity while topspin acts with gravity to drive balls down. Sidespin balls curve sideways - I will leave it to you to work out which balls curve where. Importantly topspin balls tend to bounce higher even though the interaction of spin with the court should make them bounce lower. The effect of aerodynamics on the trajectory of the topspin ball - diving it down towards the court - overrides other effects. Similarly, slice balls with their flatter trajectory tend to bounce lower. I like to analogize a sliced ball's path as straight like a laser beam - which is great at the net or midcourt but tricky from the baseline. Conversely, topspin is great from the baseline, where the goal is to hit safely high over the net and still have the ball drop in, but it is useless at the net.

How does one achieve maximum racket head speed? With .25 seconds to accelerate the racket head from zero to 78 mph the average inertia force of the racket is 15 pounds (see above), so you must apply a significant force through the arm during that brief period. The question here revolves around the most efficient way to transfer the force of the muscles to the racket head to create racket head speed. The answer is quite simple; lag and extend.

The Lag Phase

It comprises a lever arm with a pivot (or axle or fulcrum) located nearer to the counterweight which supplies the torque which accelerates the projectile at the other end. With the driving force and fulcrum at one end, applying a strong force over a short distance can result in maximum acceleration on the other end of the lever. To enhance energy transfer the trebuchet carries its projectile is in a pouch at the end of a long sling attached to the end of the lever arm. The sling acts like the tennis racket, and the attachment of the sling to the lever acts like the wrist joint. When the counterweight is released, the projectile is dragged behind the end of the lever until the counterweight reaches the bottom of its swing and stops accelerating the lever. At that point, the projectile and pouch swing around the end of the lever until they reach a point where the projectile is released. The trebuchet can, therefore, transfer energy in two ways; by accelerating the projectile around the axle during the period of maximum torque, and by whipping the sling around the hinge as the counterweight slows down. This second push on the projectile, or 'slingshot effect,' adds momentum and speed to the projectile. As a side benefit, dumping more momentum and energy into the projectile helps to slow the arm of the trebuchet down gracefully and efficiently.

The point of the lag in tennis is to shorten the lever arm early in the acceleration phase of the stroke and lengthen it later. Early in acceleration, the racket's inertia makes the head of the racket heavy. Remember that we are actually swinging the racket head on the end of a stick that comprises our hitting arm and the shaft of the racket We are sending the racket head into orbit around our body. Since we have to accelerate the racket head around a circle the difficulty getting the racket up to speed depends not just on the mass of the racket head, but also on how far it is from the muscles that are trying to drive it. Imagine trying to fling a pumpkin with a shovel. The longer the shovel, the harder it would be to get the gourd going. Once an object is moving, accelerating it seems to require less force, but now the problem becomes keeping up with it so you can continue to accelerate it. I guess that's why bobsled pushers need to be able to run fast, not just push hard. Ideally, it would be great to have a shovel that started short then got longer throughout the throw. Since your 'shovel,' aka your hitting arm, is divided into sub-levers (upper arm, forearm, and racket) that are controlled by muscles, you can change the length of the lever arm during the stroke and achieve the same effect. Lagging is one way to accomplish this.

Ideally when lagging there should be a 90-degree angle between the forearm and the shaft of the racket One way to optimize the lag is through the grip. Some grips and some grip variations make it easier to lay the wrist back into the lag position. The continental grip for both the serve and two-handed backhand and the "western" or semi-western grips for the topspin forehand optimize lagging and can greatly increase leverage.

Extension

Along the same lines as lagging is extension. Extension is simply starting a stroke with a bent elbow but straightening the arm before the moment of contact.

Extension is so pathetically simple one may wonder why I am bringing it up; it is natural to extend into the ball. The real issue is understanding why we do it. I always imagined that I was somehow using my triceps muscle to add juice to the ball. For the most part, this is an innocent and harmless misapprehension, but it can lead to dark places and untended consequences. In the serve, for example, the idea that you are hitting the ball with your triceps can be disastrous for both power and control. Forcibly driving the racket through the ball with the triceps erases the effects of the stored control and spin forces because the triceps muscle is stronger than the pronators of the forearm. Instead, we need to focus on the real role of extension; which is the appropriate and timely lengthening of the hitting arm to optimize leverage. That in mind, we know that elbow extension in the serve should start early, end early and be all done before we make contact with the ball. It should enhance snap, not erase it. It should also be easy and passive, not forced. During the lag phase of every stroke the racket is orbiting the body, and so there is centrifugal force tending to extend the elbow anyway. We are just allowing the elbow to extend by not opposing centrifugal force with our biceps. The benefits are a more efficient use of the power wave and, as a side effect, improved contact with the ball. It is easier to address the ball in extension since a wider orbit of the racket means two things; slower rotational velocity of the racket face making contact easier to time despite a larger linear velocity vector in the direction of the opposite court for optimal pace. Yes, that is a mouthful of physics gobbly goop, but consider this: As a figure skaters' arms go out the skater's rotation slows, but because of the distance from the center of rotation their hands actually travel faster.

Counter-rotation

The principal source of momentum for the power wave comes from setting ones feet and pushing off against the Earth. Should you fail to achieve adequate traction, then counter-rotation is your best source of power for pace, spin, and control. Counter-rotation depends on Newton's third law; "For every action, there is an equal and opposite reaction." If you send your left arm into orbit across the front of your body, your body and right arm will rotate in the opposite direction - kind of like a forehand. If you throw your left arm behind you, your body and right arm rotate in the direction of a backhand. If you observe the pros, especially when they volley, they do this all of the time even when they set their feet and push off. Counter-rotation adds power for pace and assures that there will be at least some rotational momentum to power the load phase of the stroke - the source of stored control and spin forces. Also, the back leg can be used to generate rotational power. If you throw your back leg behind your front leg before the lag phase, you can generate useful power. Conveniently this is a symmetric situation, with the back leg going behind the front leg in forehand, backhand and serve, so it is pretty easy to remember even for someone as body-memory-challenged as myself.