Velocity of a Golf Ball after collision


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Intuition tells us that the larger the velocity of the club head at impact with the golf ball, the larger the velocity of the golf ball after the impact. To calculate the velocity of the golf ball we need to consider the relationship between the velocity of the club head to the velocity of the golf ball.

If the collision between club head and ball were elastic we would be able to use Conservation of Mechanical Energy and Conservation of Momentum to determine final velocities of club head and ball after collision, but the golf ball undergoes some deformation at time of impact, thus some energy is lost.

"http://dustbunny.physics.indiana.edu/~dzierba/hp2000/FermiProbs
/golfball.jpg" 		A simple test of bouncing a golf ball on a solid surface shows that the golf ball does not return to it's initial height. The elasticity of a ball (e) is equal to the proportion of the velocity before collision to the velocity after collision. If the collision was elastic, e = 1. If the collision was perfectly inelastic, e = 0. The value of e is between 0.70 and 0.80.

In order for there to be a collision the initial velocity of the club head must be greater than the initial velocity of the golf ball. Also for there to be separation, the final velocity of the golf ball must be greater than the final velocity of the club head. Thus the equation for elasticity (e) of the golf ball.

1) e = (vf - Vf) / (Vi - vi)
where V is the velocity of the club head, and v is the velocity of the golf ball.

Using the principle of conservation of momentum:
2) MVi + mvi = MVf + mvf
where M is the mass of the club head, and m is the mass of the golf ball.

The unknowns in the equations will be the final velocities of the golf ball and club head, so first eliminate one of the final velocities in equation 1 and substitute into equation 2.

3) Vf = vf - e(Vi - vi) sub into equation 2
4) MVi + mvi = M[vf - e(Vi - vi)] + mvf

Since the golf ball is originally at rest, vi = 0, solving for final velocity of golf ball yields:

5) vf = [MVi(1 + e)] / (M + m)

Using similar algebra steps to solve for the final velocity of the club head yields:

6) Vf = Vi(M - me) / (M + m)

You can use these results and the fact that potential energy at the collision point is equal to zero to calculate the energy lost in the collision.

7) W = -(0.5MVf2 + 0.5mvf2 - 0.5MVi2)

Click on Tiger Woods to see an example of the velocity of a Tiger Woods drive and the energy lost to deformation of the ball.

 
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