The World's Most Efficient Tennis Racquet ... Sort Of
By Rod Cross
On May 17 this year, I built and tested the world’s most energy-efficient tennis racquet. It’s 100 percent efficient. Maybe only 99 percent, but inventors are allowed to brag a bit. All of the energy I generate during the swing goes into my arm and racquet. When I hit the ball with this new racquet, ALL of that energy goes into the ball, and NONE of the energy is left over in my arm or in my racquet. My arm stops dead on impact and so does the racquet, while the ball comes off the racquet with ALL of the energy that was previously in my arm and racquet. How good is that?
The Follow-Through ‘Problem’
The problem with all previous racquets ever built is the “follow-through.” When a player hits the ball as fast as possible, the arm follows through and so does the racquet. The ball picks up a certain amount of energy from the racquet, but a lot of the energy generated in the original wind-up and swing is retained in the player’s arm and in the racquet. That’s a huge waste of energy.
A lot of work is done when a player winds up to send down a fast first serve. The legs push up on the body, the hips rotate the trunk, the trunk rotates the shoulder, the shoulder rotates the upper arm, the upper arm rotates the forearm, the forearm rotates the racquet, then the wrist gives the racquet a final flick just before it wacks the ball. At each step along the chain, the racquet builds up speed at the expense of the previous links in the chain. Energy gets transferred along the chain as each segment comes to a stop and transfers its energy to the next segment. Just before impact, all of the energy is in the forearm and the racquet, because the hips, torso and upper arm have all finished their job and have stopped rotating. What happens next is a disaster in terms of the flow of energy. The racquet hits the ball, the ball picks up a bit of energy out of the racquet, and then comes the follow-through. Someone who knew nothing about tennis might think that the primary objective was to smash the racquet into the shins as fast as possible.
Suppose one designed a racquet that completed the chain of events in the following way: Just before the racquet hits the ball, the upper arm comes to a complete stop and transfers all its energy to the forearm. Then the forearm comes to a complete stop and transfers all its energy to the racquet. Then the racquet comes to a complete stop and transfers all its energy to the ball. That way, there is no energy wasted. It all ends up in the ball. That is exactly how I managed to construct the world’s most energy-efficient tennis racquet.
How Was It Done? Getting all the energy out of the racquet into the ball
The first problem was to get all the energy out of a racquet and into a ball. That was the easy part. I did that a few years ago and have a working model on my desktop as an executive toy. The racquet swings like a pendulum about an axis through the handle until it strikes a ball hanging in its way. The racquet comes to a complete stop and transfers all its energy to the ball. The ball swings away from the racquet, but it is mounted on a piece of string so it swings back onto the racquet and transfers all its energy back to the racquet. You can buy something similar in executive toy shops where a row of balls suspended on string strike each other. It’s called Newton’s Cradle.
The balls in Newton’s cradle are all the same mass. That’s the secret of how it works the way it does. If one ball was heavier than the other, and the heavy ball was swung to one side, the heavy ball would “follow through” after it collided with a light ball. My racquet and ball version works the same way, provided the effective mass of the racquet at the impact point is equal to the mass of the ball. The effective mass is not the same as the actual mass. Effective mass is less than actual mass, especially near the tip of a racquet where the effective mass of a real racquet is about the same as the actual mass of a real ball (57 gm) (See The Physics and Technology of Tennis for further explanation).
Getting all the energy out of the arm into the racquet
The next, and biggest, problem in designing the perfect racquet is getting all of the energy out of the arm and into the racquet. One part of this problem is getting a good match between the player’s arm and the weight and swingweight of the racquet. Another problem is getting the player to swing the racquet using the right technique. This problem has been examined in regards to the golf swing. A golf club is swung in almost exactly the same way as a double pendulum. A simple pendulum has a fixed pivot point at the top and a mass that swings to and fro underneath. If you attach another simple pendulum to the bottom of the first one, you have a double pendulum. A golf club acts like the pendulum at the bottom and the player’s arms act like the pendulum at the top. Instead of swinging to-and-fro in a regular fashion, a golfer just has one giant swing starting with the club above the head and ending with the club striking a ball.
One might think that as the club hits the ball, both the club and the player’s arms are swinging at maximum possible speed. In fact, what actually happens is that the arms slow down as the club picks up speed. The arms don’t come to a complete stop but they definitely slow down before the club hits the ball. It’s not that the player deliberately does this is or is even aware that he or she is doing it. The fact is, the club rotates at such a high speed that it pushes backwards on the hands and forces the arms to slow down. The same thing happens when a batter swings a bat or a tennis player swings a racquet or when a double pendulum is pulled aside and released.
The physics of all this are very tricky. It helps to imagine that the motion of the racquet is due to two separate parts. One part is pure rotation of the racquet about its balance point. That is the part that causes the handle to push backward on the hand and slow down the forearm. The other part of the motion is its forward motion. The racquet doesn’t just spin in mid air around its balance point. It also accelerates forward because the player pushes it forward. Newton’s third law of tennis says that the handle will push back on the player, but the main push backward arises from high-speed rotation of the racquet.
I analyzed a high-speed, slow-motion video of a top college player serving at top speed. Just before impact his racquet reaches top speed and his forearm slows down. The forearm slows down just before impact but it doesn’t come to a complete stop before impact.
I therefore made a simple double pendulum out of two aluminum bars to investigate a bit further. I used a digital video camera to record the results and found out three things. First, if the bottom pendulum is about half the weight of the top pendulum, the slowing down effect is so strong that the top pendulum not only comes to a stop, but it reverses direction while the bottom pendulum is still gathering speed. Second, if the top pendulum is about six times heavier than the bottom pendulum, then the top pendulum slows down but it doesn’t come to a stop. And third, if the top pendulum is about four times heavier than the bottom pendulum, then the top pendulum comes to a complete stop just as the bottom pendulum reaches the bottom of its swing where it is travelling at maximum speed.
We have here a solution to the follow-through problem. If the racquet is too light, the forearm slows down a bit before impact but it doesn’t come to a stop during the swing and it follows through after the racquet strikes the ball. If the racquet is heavy enough then the forearm will come to a complete stop just as the racquet is about to strike the ball, in which case all of the energy generated by the player is in the racquet and there is nothing remaining in the player’s arm (Figure 1E). At that point, the racquet strikes the ball and all of the energy in the racquet is transferred to the ball if the impact point is just right.
Back to reality
There are a couple of hitches in all of this. An adult forearm has a mass of about 1600 gm. In order to get all the energy out of the forearm into the racquet, one needs a racquet mass of about 400 gm, in which case the ball needs to be around 100 gm. Currently, balls are about 57 gm. That’s a problem unless the rules change.
The other hitch is that even if a 100 gm ball was allowed, it would end up travelling at a lower speed than a 57 gm ball, despite having more energy than the lighter ball. A 100 gm ball travelling at say 40 m/s has a kinetic energy of 80.0 Joules. A 57 gm ball travelling at 40 m/s has a kinetic energy of 45.6 Joules. If two balls are travelling at the same speed, the heavier ball has more energy than the light ball. The kinetic energy of a ball is calculated by taking half its mass and multiplying by its speed squared. There is therefore no guarantee that a heavier ball with more energy will travel any faster than a light ball with less energy. Even if a 100 percent energy-efficient racquet can be constructed, the extra energy in the ball that results will be gained at the expense of a decrease in ball speed. This is what happens whenever a heavy object collides with a light object. If the light object is made a bit heavier, it will gain more energy during the collision but it will end up travelling slower.
So, where does that leave us? It leaves us with three options. You can use a heavier ball if you want more energy in the ball, but the ball will come off the racquet at a lower speed. Or you can use a lighter ball if you want the ball to come off the racquet faster, but it will have less energy. Or you can stick with what you’ve got, which is about as good as it gets.
See all articles by Rod Cross
About the Author
Rod Cross retired in 2003 as an honorary member of the Sydney University staff and continues to work on the physics of sport and forensic physics. He is a physicist and co-author of The Physics and Technology of Tennis available from the USRSA.