# Table Tennis Ball Speed Calculator

By Jay Turberville
October 18, 2003

 Initial Speed (meters/second) Miles Per Hour Time Interval (seconds) Calc Iterations (precision): Kilometers Per Hour Calculated Results Final Velocity (meters/second) (kph): (mph) Average Velocity (meters/second) (kph): (mph) Distance Travelled (meters) (feet)

This calculator calculates various values based upon the initial velocity of a regulation table tennis ball (2.7 grams, 40mm diameter). This calculator only considers the drag due to air friction in its calculations.

A calculator that will allow you to use different ball sizes, weights, drag coefficients and altitudes can be found here.

The assumptions are (provided by KAGIN1 on the tabletennis.about.com forum):
D = Cd * r * V^2 * A / 2 (force of acceleration due to air resistance - Drag)
a = D / m (acceleration due to air resistance)
Cd = 0.5 (coefficient of drag for a smooth sphere, so it depends on how old the ball is)
r = 1.229 kg/m^3 (air density at sea level)
V = velocity of ball, variable
A = 0.0012566 m^2 (area of ball) (not surface area)
m = 0.0027 kg (mass of ball)

When simplified this yields:

a = 0.143 V^2

If Velocity = a * time, then the change in velocity (if negative) would be: Vsubsequent = Vinitial - [(0.143 * Vinitial2) * time]

At this point, things seem to get tricky. The force of accelleration (decelleration if you must) due to air friction is continuous. If you perform a single calculation and use a time interval such as a half of a second, you will get a very wrong result. This is because the ball's velocity change continuously over that one half second. But if you make ten calculations at one twentieth of a second intervals and apply the resulting velocity change to each subsequent calculation, you will reduce the error substantially. Smaller time divisions and a greater number of calculations will result in more accurate results. This is the approach KAGIN1 suggested and is the scheme used in this calculator. The calculator defaults to 1000 calculation iterations (and alters the time interval for the number of calculations accordingly), but you can change it upward or downward to see how it affects your results. 1000 seems to be a good compromise between speed and accuracy.

To use the calculator, you need to enter the ball's initial velocity and the length of time it is travelling. Velocity needs to be entered in meters per second. Time will typically be in fractions of a second (decimal values). The calculator will determine average velocity, distance travelled and final velocity and will provide these numbers in multiple units of measure.

The calculator includes a converter so that you can easily use miles per hour or kilometers per hour. Just enter the mph or kph value, click the appropriat "convert" button, and the calculator will convert to meters per second and put that value into the form.
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