# Q & A: tennis racquet speed

Q:
There is a tennis player who can hit a 146 mph serve. Is there a way to determine how hard the ball would have to be hit to gain that kind of speed??
- Ashley
Wisconsin
A:
I guess by how hard youre asking what the racquet speed has to be when it hits the ball. I can give a minimum estimate. Lets say that the racquet has a lot more mass than the ball and that the collision loses no energy to sound and heat etc. the first approximation is pretty good, the second not so good. Both let us underestimate the speed needed.
Looking at the collision in the frame of reference where the ball is moving and the racquet standing still makes calculations easy. The way the collsion can conserve energy in that frame is for the ball to simply switch directions from moving toward the racquet to moving away. So the change in velocity of the ball is twice its velocity relative to the racquet.

In the frame of the tennis court, where the ball is initially at  rest, the racquet must then initially be moving at least 73 mph. I wouldnt be surprised if the real number were more like 100 mph, given our approximations.

Mike W.

Only the part of the racquet hitting the ball needs to move that fast.  The handle can be moving a lot more slowly if the racquet is rotating while moving towards the ball, which is usually the case.

Tom

(published on 10/22/2007)

## Follow-Up #1: Followup on Tennis Racquet

Q:
Would you provide the formula you used to derive your answer? We have been arguing the difference between e=mcsquared and f=mc. One of us says e=mc squared best estimates, simply.
- george pulver
naples florida
A:
Where did those formulas come from? E=mc2?  f=mc doesnt even sound like it makes sense. I don't see how c, the speed of light,  got involved in a tennis question.
All I used was:

(change in ball velocity)= twice (difference in velocity between ball and racquet).

That formula would apply if the assumptions (elastic collision, very massive racquet) were true- an approximation. Its a simple mathematical consequence of combined conservation of energy and momentum. We start with basic principles like that that, not some formula. Since the ball is initially almost at rest with respect to the ground, this becomes simply final ball velocity = twice initial racqet velocity.

Mike W.

(published on 10/22/2007)