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The Basic Physics and Mathematics of Table Tennis / Ping-Pong

2 Bats + 1 Ball + 1 Net + 1 Table + 2 Players = Lots of Fun!!


My thanks to guest author Jonathan Roberts, who has kindly taken the time to write about the physics of table tennis, saving me the need to strain my brain trying to figure this stuff out!

Firstly, a very brief introduction to the mathematics that is used to describe Table Tennis. There are a handful of formulae that are used, which a man called Sir Isaac Newton derived in his monumental work Philosophae Naturalis Principia Mathematica. Incidentally, this work is generally regarded as the single most important work ever written in the history of science, and I regard Newton as the greatest scientist to have ever lived.

It accurately explains how objects move from the scale of interstellar objects (galaxies, stars, planets, SERIOUSLY BIG STUFF etc.), down to things on the scale of about 1000th of a millimeter or 1 micron. After that, this model of the universe starts to break down and you need to go to Quantum Theory and Relativity, which involves FRIGHTENING Mathematics and Physics to use.

Anyway, this is the Physics and Mathematics of Table Tennis in the Newtonian Universe.

The basic formulae to be used here are:
P = W÷t
W = Fs
F = ma
a = (v - u)÷t Note: This is usually rearranged to v = u at
T = rF
Note: When two letters are next to each other it means multiplication. This is the correct notation. Take the second formula as an example, W = Fs This is expressed as W = F multiplied by s or W = F x s.

P = Power (The amount of oomph that is applied)
W = Work (The amount of energy that is consumed)
t = Time (Length of time the Power is applied for)
F = Force (Basically the amount of grunt the shot has. Similar to P but subtly different)
s = Displacement (this essentially translates to Distance, except under certain circumstances)
m = Mass (weight of the ball, fixed at 2.7g)
a = Acceleration (change in velocity over a given time period)
v = Velocity (speed of the shot)
u = Initial Velocity (how fast the ball is hit at you)
T = Torque (The amount of Turning Force that is applied)
r = Radius (the length from the middle of a circle, to the perimeter.)

P = W÷t
In order to gain more power in your shots, you have to do more work or take less time in your shots. The time in a shot refers to the time the ball is in contact with the racket which is fixed at approximately 0.003 seconds. Therefore, in order to increase the Work done, the second equation must be examined:

W = Fs
If the amount of Force is increased, then the Work coefficient is increased. The other way is to increase the Displacement, but that cannot be done as the length of the Table is fixed (technically, lobbing or looping the ball will increase the Work done, as the ball has to cover a larger distance than ball that barely clears the net). In order to increase Force, the third equation must be examined.

F = ma
In order to increase the Force, the Mass of the ball needs to be increased which is impossible, or the Acceleration needs to be increased. In order to increase the acceleration, we analyse the fifth equation.

a = (v - u)÷t
The result of the calculation between the brackets must be calculated first (it's a mathematical law). Therefore you want to maximize the acceleration, minimize the initial velocity. In order to maximize the velocity, you have to hit the ball as hard as you can. The initial velocity is something you have no control over, as it is how hard the opposition hits the ball at you. However, as the initial velocity is coming towards you, its value is negative. So it is actually added to your velocity, as subtracting a negative number actually means you add the two terms (another mathematical law). The time remains fixed, for the reason explained above.

Therefore this demonstrates why the harder you hit the ball, the more Power it will have.

But, speed is not everything in Table Tennis. There is spin, which will now be discussed.

All About Spin

Jonathan discusses the subject of spin in table tennis here. Read this before reading the text below.

Reaction Speed in Table Tennis

From a biological perspective, there are limits to how fast the body can react to a stimulus. There is a difference in this time between an audio stimulus and visual stimulus. Technically we respond faster to an audio stimulus than a visual stimulus, 0.14 of a second compared with 0.18 of a second respectively. Therefore, if you can work out EVERYTHING about the shot you need to just by hearing it strike the racket, you're 0.04 or four one hundredths of a second faster than anyone else who has ever played table tennis before.

Good players (even average players like myself) can still deduce a lot of what the opposition is doing, simply by listening to the noise the ball makes when it contacts the bat. For instance a brushing noise of the ball on the bat tells you that spin has been put on the ball, hitting a loop will give this effect. A sharper 'pock' will tell you that the ball has been struck quite solidly, and will also tell you that they're using a thin rubber. It is, of course, legal to ask to view the opposition's bat, so listening to the noise to tell what thickness rubber is being used is just something that can be done.

Some people say that when the ball strikes the table they can tell whether the ball is top spun or under spun. Personally, I can't, but it wouldn't surprise me that elite players can.

In Table Tennis, the average total time to react to a shot is usually around 0.25 of a second. With lots of training and lots of practice, this can be reduced to 0.18 of a second. This is one of the big factors in what separates the greats of table tennis, from the top A grade players. In elite levels of the sport, even being the smallest fraction of a second (1/1000ths) faster starts to make a difference.

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