# Ball Speed Calculator

### Calculate ball speed changes due to air drag.

By Jay Turberville
October 18, 2003

 Drag Coefficient (default 0.5 - sphere) Elevation (feet) Air Density (kg/m3) Elevation (meters) Ball Mass (kg) Ball Diameter (meters)

 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 is built on the core of the Table Tennis Ball Calculator and starts with base variables matching that calculator. But this calculator adds a number of variables so that different ball sizes, weights and drag coefficients can be used as well. It also allows for adjustments based on altitude.

Altitude values must be entered in feet. I have, however, provided a converter so that you can easily convert meter values to feet. To change altitude, simply enter a value for the altitude in feet (or enter a meters value and click the "convert meters to feet" button) and then click the "Calculate Air Density" button. If you calculate the air density at zero elevation, you will get a slightly different value than the default air density. This is a correct result. I've simply kept the same air density default value that I used in the original Table Tennis Ball Calculator for consistency with that calculator. It is a trivial difference.

For other conversions, the Science Made Simple site offers a number of conversion calculators.

I have a minimal understanding of the mathematics behind the air density calculation. I simply looked at the structure of Ilan Kroo's Standard Atmosphere Computations calculator and copied his method.

This calculator will calculate various values based upon the initial velocity of a sphere shaped object or any object with a known coefficient of drag. This calculator only considers the drag due to air friction.

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]

The 0.143 constant value changes (though it is not displayed) if the ball diameter, ball mass, coefficient of drag, or air density default values are changed. The force of accelleration (decelleration if you must) due to air friction is continuous. If perform a single calculation and use a time interval such as a half of a second, you will get a very wrong result. 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 the 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, but 100,000 or even 1 million iterations are processed fairly quickly on 1Ghz PCs.

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.

 Ball diameter (meters) mass (Kilograms) 38 mm TT ball 0.038 0.0025 baseball 0.074787 0.148835 tennis ball 0.065 0.0577 golf ball 0.04267 0.04593

Be aware that textures on the surface of a ball can significantly decrease the drag coefficient at high speeds. Baseball seams, golf ball dimples and tennis ball fuzz all have this effect. So at typical speeds, they can exhibit drag coefficients that are half that of a smooth sphere. Read this article for more information.