Author: Yang Baisong | University of Chinese Academy of Sciences
Training unit: Institute of Physics, Chinese Academy of Sciences
Review: Wang Yumei | Wang Yumei, Institute of Physics, Chinese Academy of Sciences
Tennis is a very enjoyable sport, the agile movement of athletes, the violent collision between the racket and the ball, the quiet grass, the unrestrained red clay, a tennis game can really be described as an audio-visual feast. I believe that many people have followed tennis and dreamed of galloping on the tennis court like Federer, Nadal, and Djokovic. However, when everyone came to the tennis court full of confidence and anticipation, they bought their favorite tennis rackets and a lot of tennis equipment, and found that things were not so simple! Obviously, the athletes on TV can play wonderful rounds, but when it comes to themselves, they can only serve, and they can't catch a single ball. Looking at the opponent again, it seems that he is playing worse than himself, and he secretly thinks that tennis is really difficult, and from then on, the tennis racket bought at a large price can only eat dust in the corner. If you've had this experience, I'm sorry for it. If you had met the editor earlier to tell you about tennis physics, maybe you would have become a master of one in a hundred by now.
Why do the balls keep flying?
When it comes to tennis, the first thing that comes to everyone's mind is whether it is a perfect parabola like the following, but in fact it is not, because tennis is not only affected by gravity when moving, but also by air resistance, buoyancy and Magnus force (you may have heard of it for the first time, but I believe the editor is not so difficult to understand). When the ball speed is not very fast and there is no spin, you can ignore the influence of these forces and think that the tennis ball is only affected by gravity, so that the ball is a parabola.
But is this kind of curved ball really a good ball?
In this case, the trajectory of the ball is a parabola, but it is not perfect, because you will find that the ball is not attached to the net, but flies very high, and the landing point is not controlled by yourself at all. If that's the case, it's because you're missing a secret weapon – spinning. Students who have played table tennis should be familiar with this. With this magic weapon, you can play as hard as you want without worrying about the ball flying out of bounds, and you don't have to play carefully anymore. Let's take a look at why spin is so important for playing tennis.
How does a tennis ball spin?
The movement of tennis begins with the force exerted by the racket on the tennis ball, which can be simply divided into three situations according to the relative position of the racket and the ball:
- When the force exerted by the racket to the tennis ball exceeds its center of ball, the object is only pushed by one pass through the center of the ball, as shown in Figure A below.
- When used as a force shown by the blue arrow in Figure B below, according to the principle of equivalence of force, the effect is equivalent to adding a pair of equilibrium forces of the same magnitude and direction to the center of the sphere that are the same as the force. In this way, the tennis ball is acted upon by a push force (red arrows) and a pair of force couples (green arrows and blue arrows). This coupling is one of the sources of tennis topspin.
- The same tennis ball in diagram c below will produce a downspin.
In addition, the spin of the tennis ball can also be obtained by moving the racket up or down.
- As shown in Figure A below, when the racket only moves forward, the tennis ball only receives the force of the center of the ball, and the ball is a flat shot.
- As shown in Figure B below, when the racket has an upward motion relative to the ball, in addition to the force of the tennis ball passing through the center of the ball, an additional moment will be applied to the tennis ball, which will cause the tennis ball to spin up.
- As shown in Figure C below, when the racket moves downward relative to the tennis ball, the tennis ball produces downspin accordingly.
Both of these methods will give the tennis ball spin, so now you know how to get the maximum spin on the ball we hit? The answer is a combination of the two. When we want to play topspin, we tilt the racket forward, and at the same time the racket moves upward, and there is a feeling of brushing the ball upwards (as shown in Figure B below), then the result is a very powerful topspin ball. If you want to play a downspin ball, tilt the racket back and make a chopping motion (as shown in Figure C below), then you will hit a high-quality downspin ball.
What are the factors that affect tennis?
The movement of a tennis ball in the air with a large force and rotation cannot be considered to be affected only by gravity, and the Magnus effect must be taken into account. The so-called Magnus effect means that when the direction of rotation of the tennis ball does not coincide with the direction of its flight speed, a transverse force will be generated in the direction perpendicular to the plane composed of the rotational angular velocity vector and the translational velocity vector, and the phenomenon that the object's flight trajectory is deflected under the action of this transverse force, which is also known as the Magnus force. The rotational angular velocity vector, the translational velocity vector and the Magnus force satisfy the left-handed spiral rule.
The above explanation of the Magnus effect is still a bit too awkward, in fact, the effect can also be intuitively explained with the knowledge of junior high school physics, everyone must have learned Bernoulli's theorem: the pressure is small where the air velocity is large, and the pressure is strong where the air velocity is small, so a transverse force will be generated. So for the tennis ball, if it has the topspin and moves to the right as shown in the figure below, then the tennis ball will be subject to the air resistance to the left, because the air is viscous, the tennis ball with the topspin will drag the air above it to slow down and the air below the ball will be accelerated, which makes the air flow velocity above the ball small, the pressure is strong, and the air flow velocity of the air below is large, the pressure is small, the final effect is that the tennis ball will be subjected to downward force, that is, the Magnus force mentioned above. Similarly, if the tennis ball has a downspin, the Magnus force will be perpendicular to its direction of motion upwards.
The Magnus force is proportional to the speed of the tennis ball, proportional to the speed of the tennis ball, and proportional to the cube of the diameter of the object, so the larger and faster the volume and speed of the tennis ball flying upwards, the greater the Magnus force on it. The Magnus force on a sphere can be expressed by the following formula:
where is the Magnus force, A is a coefficient, which is related to the surface coefficient of the ball and the internal friction and viscosity of the air flow, ρ is the density of the air, d is the diameter of the tennis ball, v is the rotation speed of the tennis ball, and w is the motion speed of the tennis ball.
So what is the actual trajectory of a ball with rotation?
The true trajectory of tennis motion is the result of a combination of gravity, air resistance, buoyancy and Magnus forces. If the tennis ball is topspin, the velocity of the tennis ball in the vertical direction will slowly decrease to 0 due to the action of the Magnus force and its own gravity in the vertical direction of the ball during the rising phase of the ball, so that the trajectory produces an inflection point, and before the inflection point, the horizontal component of the Magnus force is the same as the horizontal motion of the tennis ball. After crossing the inflection point, the horizontal component of the Magnus force on the tennis ball is opposite to the horizontal motion of the tennis ball, and its horizontal velocity will rapidly decrease and cause the tennis ball to fall suddenly. So when the tennis ball is topspin, there is no need to worry about using too much force to get the tennis ball out of bounds, the greater the speed of the ball, the greater the Magnus force on it, and the ball will always fall steadily into bounds.
The same principle, when the tennis ball has downspin, the Magnus force is always upward, although the force we exert to the ball when playing the spin ball is relatively small, the speed of the ball is not fast, but the overall downward force is small, the ball is easily pushed to a position close to the bottom line, but in order to control the downspin ball in the boundary, you also need to practice the feel of it.
Why is a spinning ball so threatening?
In fact, the magic of the topspin ball is not only to be able to control the ball in bounds, do you feel the strong impact of the opponent's topspin ball after the rebound when you play with the master, and it feels like it has been re-accelerated. And so it is. The re-bounced ball was indeed accelerated twice, why is this happening?
First of all, when the topspin ball is in contact with the ground, it moves backwards relative to the ground, and the ground will exert a forward friction force on the ball, which directly leads to the increase of the speed of the ball in the horizontal direction, and the Magnus force and gravity of the ball after bounce are downward, and the speed of the ball in the vertical direction decreases. The combined effect makes the ball lower and faster, producing a strong impact, which is also known as the secondary acceleration of tennis.
So in the end our topspin will move like this
The horizontal velocity of the ball decreases and the height of the bounce increases, as if the ball floats over and suddenly stops. The complete trajectory of the movement is shown below
After some explanations by the editor, are you now full of confidence and want to pick up a tennis racket to practice it immediately? Here I will tell you some practical skills. If the opponent's return ball is slow and has little power, then threaten the opponent with a topspin return. When the opponent's ball is very fast and you don't have time to make adequate preparations, use the downspin ball to transition and try to return the ball to the opponent's baseline to buy yourself time to prepare. If you want to learn tennis well, you still need to practice more and master the correct hitting movements, so that you can apply it freely when playing.
I sincerely hope that students who love tennis can truly appreciate the fun brought by tennis and savor the beauty brought by sports in addition to study and work.
bibliography[1] Ma Dahui. Aerodynamic analysis of the flight trajectory of a tennis ball topspin ball [J]. Sports Research and Education, 2011, 26(5): 115-118.
[2] Pan Huiju. Mechanical model of Magnus effect [J]. Zhejiang Sports Science, 1995, (3): 16-19.
Editor: Mu Zi