1 The Physics and Math of Ping-pong and How It Affects Game Play 1 The Physics and Math of Ping-pong and How It Affects Game Play By: Connor Thompson & Andrew Johnson The Practical Applications of Advanced Mathematics Amy Goodrum, Anna Hunt July 15, 2010
2 The Physics and Math of Ping-pong and How It Affects Game Play 2 Abstract Table tennis, or ping-pong, has been played for centuries, dating back to Asian cultures until becoming very popular in England and throughout Europe and the world. The game, since its beginnings, has become more complex as its popularity grew, having more rules, new ways to hit it, and different equipment. Most people understand ping-pong as hitting a ball back and forth between two people until someone misses; however using physics has become a large part of playing and becoming skilled at pingpong. Our goal in our research was to study the possible physics that lie in the background of the game and experiment to show what serve, hit, and the angle of hitting the ball gives the player the highest advantages. There are three major concepts of physics we studied and discussed in our paper: aerodynamics (dealing with spinning the ball), force (dealing with the power used to hit the ball), and projection (dealing with the path the ball goes if you use a specific angle to hit it). After experimenting with the types of spin (top-spin, down-spin, and side-spin), the angles (90, 75, 60, 45, 30, and 15 degrees, either under handed or over handed), and the force used (relatively speaking, light, medium, or heavy force) we came to the conclusion that the most difficult to return (while still being relatively easy to serve) spin was down-spin, the most difficult angle was about degrees, and the most difficult amount of force was between medium and heavy. Based on that and other data, we concluded that the best (meaning hardest to return while being easy to hit/serve) hit was the backhand backspin at slight heavy force.
3 The Physics and Math of Ping-pong and How It Affects Game Play 3 Background Table tennis or Ping-pong has evolved from a primitive game played solely for fun to an Olympic sport played all over the world. Ping-Pong started in England with a game called whiff-whaff that is very similar to today s table tennis, except without the table. All you needed was a table, a bat or paddle, a net and a round ball. The first two pictures in the appendix show how the game has evolved. In 1905 the game became very popular in England but it was not until 1920 that the game got recognition all over Europe. It was in 1920 that many national associations began and official rules made (The History of Table Tennis, n.d.; Table Tennis, n.d.). The simplicity of the game, table tennis, is surprising for its rapid rise in popularity. The game today is played on a table 2.74 meters (9 feet) long and meters (5 feet) wide. The table rises.76 meters (2.49 feet) above the ground. The net is in the middle of the table separating the table into two distinct halves. The net itself is.15 meters (6 inches) high and stretches the entire length of the playing surface. The paddles must have a red and black side (if it is an official game because of the ITTF s official standards). Last, but certainly not least, is the ball, which is spherical in shape, has an international standard size of 40 millimeters in diameter, and must be either white or orange (Official Rules of Table Tennis/Ping-Pong, 2010; Table Tennis Strokes, Drills, and Selecting Equipment, 2000). Table tennis has fairly simple rules that are easy to understand. A match is usually played with the winner winning three of five games (though it can be four of
4 The Physics and Math of Ping-pong and How It Affects Game Play 4 seven, five of nine, or so on) and to win a game you must be the first player to reach eleven points. However, if the player to reach eleven is not ahead by two or more points, the game continues and the winner is the first player to be ahead by two points. Once a player has succeeded in getting 11 points and won by two, that game is over and there is a break. Then they will continue on the opposite side of the table. To starting a volley, the serving player must complete a serve. To serve the ball, a player would take the ball on his open palm, throw it up at least six inches, and then strike it so it hits his side once and bounces to the opponent s side; if the ball hits the net then the other side of the table, it is a let serve and the player must re-serve with no score counted, but if it hits the net and comes back to the player or goes off the edge of the table then it is a point for the player who was supposed to receive the serve. Each player serves twice in a row before it rotates and if it is a game point, meaning a point that would allow one player to win, the player with the lower score serves. The set up of the table cannot be changed, so it is illegal for a player to move the table, the net, or have their freehand touch the table (Official Rules of Table Tennis/Ping-Pong, 2010). After the serve is completed many things can happen. The volley, as it is called, is where the players keep trying to make the ball hit twice on their opponents side of the table. Obviously everyone wants to use all of the weapons they could possibly have. Some players will want to play quickly and fast using lots of power; other players want a more rhythmic game where they can place each ball where they want it to go. If this was all to Ping-pong then it would be a pretty easy game to try and learn, but spin and math must be discussed (Official Rules of Table Tennis/Ping-Pong, 2010).
5 The Physics and Math of Ping-pong and How It Affects Game Play 5 All of this can be found interesting but where is the math in all of this? Players desire to have power in ping-pong, but they are afraid they will hit the ball over the table. So, what is power in scientific terms? The simple way to put it is P=W/t. Power equals the work done divided by the time the paddle was in contact with the ball; so, to get more power you need quicker strokes of the paddle and more work done. Work is the same as the force multiplied by the distance or displacement of the ball. So in order to increase the work, you must increase the force since you can change the lengths of the ping-pong table. This brings the discussion to one of Newton s laws: force equals mass time acceleration (F=ma). Now we cannot change the mass of the ball but we can try and put more acceleration on the ball. Acceleration is basically the velocity (speed) of the ball minus the initial velocity all divided by the time (Letts, G., 2010). So, acceleration is how fast the ball is going minus what it was already going divided by the time that the paddle is in contact with the ball. So all the physics that has been discussed comes down to the velocity of the ball and velocity is simply the distance or displacement of time (d/t). The distance still may not be changed so the shorter your swing is in contact with the ball the more power you will theoretically have, but power and speed are not everything (Letts, G., 2010). People have been researching and experimenting with ways to use physics to help make their serve and game play all the more deadly, and one of the most common ways to do that is to add spin to a ball. Spinning the ball can help people make a ball
6 The Physics and Math of Ping-pong and How It Affects Game Play 6 fall quickly, bounce high, or curve in ways that make it harder for their opponent to respond and hit the ball back correctly (Letts, G., 2010). When we talk about spin we are not talking about one specific thing because there are different types. One type of spin used is top spin; Top spin refers to spinning the ball in a way that it follows its movement, so it spins forward. This type of spin is usually done at a height at or below the net height so that the ball will go over the net, sink quickly, and pop up high when it bounces. The next type of spin is down spin, which results in hitting the ball so it spins in a way that goes opposite of its movement, so it spins backward causing the ball to drop and stop or hit the table and have a low bounce so the opponent has less of a reaction time to hit it back. For this spin, the ball must be higher, because stroking it with down spin does not allow it to go up, like with top spin. The final type of spin used is called side spin, in which you stroke the ball with the paddle going completely horizontal either left or right. Side spin makes the ball go straight for a while before the spin kicks in and the ball curves either left or right. Side spin is used to either catch the opponent off-guard or to make the ball hard to hit because of the angle it comes up after it bounces (Letts, G., 2010). What is spin? Spin is basically the ball rotating on its self and this is where something called the Bernoulli Effect comes in. When you hit the ball with spin, you are rotating the ball in one direction which causes one side of the ball to move faster than the other side. This creates the Magnus Effect, which is that when you have a rotating object that it causes air motion as well. The Bernoulli Effect says that one side of the
7 The Physics and Math of Ping-pong and How It Affects Game Play 7 ping-pong ball will move faster than the other, but the Magus Effect says that air flow is also affected and this causes the real spin to be increased to the spin we see today (Johnson, C., 2003). Research Question How do you use Physics in game play? Methods To test the physics of table tennis, how the spin, force, and angle of hitting the ball affect the ball s path, we performed experiments in which we tested hitting the ball a certain way. We recorded how easy it was to serve a ball a certain way or keeping a ball in play by only hitting it that way. We tested types of spin, forehand vs. backhand, force, and the angle of hitting the ball. We used Sportcraft balls and paddles on a standard table tennis table to perform each test. In each test one person would try and perform a feat such as putting topspin on the ball while keeping the ball in play or on the table. That one person would try the same feat five times before we would switch and the other person would try the same feat five times. After that, we would add up the total balls in play and record that number. We did this five times since we had plenty to test. Results In our trials we measured different things such as amount of times we could get some spins in and return each spin. We first tried getting basic serves and in-game volleys
8 The Physics and Math of Ping-pong and How It Affects Game Play 8 in using the different types of spin and hand positions. The table below shows that we were able to consistently return and serve a ball using only a forehand topspin. We were not, however, able to keep the forehand backspin on the table as much as the topspin, but we were able to serve and return it. Out of 30 Forehand Backhand Forehand backspin topspin topspin Backhand backspin serves in-game Next we tried to use spin in two ways, one way was off of the serve the other way was off a volley. For the backhanded spins it was clear that using them off of the serve was not a good idea because the angle needed to clear the net but stay on the table was small, but when the ball was returned to us, we were able to play the ball and keep it on the table more. This is due to the ball not being bound to hit our side of the table first. By far the backhand topspin was the hardest spin to use in any position. The next test we did was using the only spin that was left out of the first test. Sidespin did prove somewhat difficult to actually use off of a serve, but once in game play it was easier to control. Using left sidespin was easier than right because the motion of the paddle was straight out away from your body at a 90 degree angle from the ball. My partner and I were able to successfully use left sidespin forehand twentyfour out of thirty times and left sidespin backhand eighteen out of thirty times. So the right sidespin provided more of a challenge because to use it we had to bring the
9 The Physics and Math of Ping-pong and How It Affects Game Play 9 paddle close to us but still maintain a 90 angle with the ball. The forehanded right sidespin was successful twenty-one times out of thirty, but the backhanded right sidespin was only successful nine times out of thirty. All of this is summarized in Appendix A. To sum this up, it is easy to go forehand with any sidespin but it is possible to get backhand sidespin if the player practices. Next, to study force and projection, we tested how different forces (low, medium, and heavy force) and different projections (either a high or low served ball) affect how hard it is for the opponent to return the serve. We combined the three forces and two projections to make six testing categories, low force and low serve (12/30 successes in game setting), low force and high serve (12/30), medium force and low serve (21/30), medium force and high serve (15/30), heavy force and low serve (21/30), and heavy force and high serve (18/30). All of this is summarized in Appendix B. The last trial that we preformed measured the amount of times we could return the ping-pong ball holding the paddle at a certain angle. We were also able to see from this test how much spin was put on each ball that was successfully returned. We started at 90 degrees and went down from there to 75 degrees. When we held the paddle at ninety degrees it was simple to return a serve. As we decreased the angle it got progressively harder to return even the most basic serve with no spin. The data from this trial is in Appendix C. We went from 75 degrees to 60 degrees then 45 degrees. After 45 degrees, we switched from an overhand hit (hitting the ball with the
10 The Physics and Math of Ping-pong and How It Affects Game Play 10 paddle moving upward) to an underhand hit (hitting the ball with the paddle moving downward. We started again at 45 degrees underhand then went to 30 degrees and then finally 15 degrees, which was by far the most difficult to return due to the precise timing need. The smaller the angle the more exact you had to be. If the ball did not hit directly in the center of the paddle when the paddle was half way through it motion, then the ball would either have to much spin and not go over the net or have too much power and no spin. So, this trial shows exactly what happens to the ball after it is hit. Conclusion If time were not a factor in our research, we could have done more in depth testing to help prevent the other limitations from affecting our results. Limitations like lack of equipment, space, or etc which prevented us from being able to get precise, quantitative data, like force, angle, and other things could ve helped us in making more conclusions and prove our current conclusions better. There are more precise instruments that we could have used, but ping-pong can be diagnosed two ways: with math equations or using your brain and seeing what makes sense. It is not that complicated of a game, but when you break it down, there can be a lot of things to think about and ideas to explore.
11 The Physics and Math of Ping-pong and How It Affects Game Play 11 References Johnson, C. (2003). Aerodynamic Lift, Bernoulli Effect, Reaction Lift. Retrieved July 9, 2010, from Letts, G. (2010). Spin In Table Tennis/ Ping-Pong. Retrieved July 12, 2010, from Letts, G. (2010). Spin in Table Tennis/ Ping-Pong - How Does it Work and How Do You Create It?. Retrieved June 27, 2010, from Letts, G. (2010). The Basic Physics and Mathematics of Table Tennis / Ping-Pong. Retrieved June 24, 2010, from (2010). Official Rules of Table Tennis/ Ping-Pong. Retrieved July 12, 2010, from (2009). Table Tennis. Retrieved June 28, 2010, from (2000). Table Tennis Strokes, Drills, and Selecting Equipment. Retrieved July 7, 2010, from (2010). The History of Table Tennis. Retrieved July 7, 2010, from The History of Table Tennis. Retrieved July 14, 2010, from
12 The Physics and Math of Ping-pong and How It Affects Game Play 12 Appendix Appendix A Number out of 30 Left Sidespin Backhand 18 Left Sidespin Forehand 24 Right Sidespin Backhand 9 Right Sidespin Forehand 21 Appendix B Force and height of serve number in out of 30 Low force and low serve 12 Low force and high serve 12 Medium force and low serve 21 Medium force and high serve 15 Heavy force and low serve 21 Heavy force and high serve 18 Appendix C Degrees Number in out of (overhand) (overhand) (overhand) (overhand) (underhand) (underhand) (underhand) 15 *For all tables, right column equals number of times serve or return was in
13 The Physics and Math of Ping-pong and How It Affects Game Play 13 Picture 1 Picture 2
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