A question about velocity and collision btw. tennis ball and bat

[EriqueSherman]

Hi. I am doing an investigation about some of the main elements of physics involved in a swedish sport called brannboll (think baseball, but instead of having a pitcher throw the ball at you, you throw it up in the air yourself prior to hitting it. Also, you use a wooden bat and a tennis ball). The practice of the sport itself is however not crucial to my problem, so you do not need to understand it fully.

Homework Statement

Ok, here is what I want to do: I want to be able to show, mathematically, how the velocity of the tennis ball before you hit it is related to the velocity of the tennis ball after it has been hit by the bat. For example: how can I show that when I hit a ball that is traveling towards me in 20 m/s, it will be returned with a higher velocity than if I hit the same ball when it comes at me in 5 m/s?
You could also think of this scenario in baseball. You would not make it travel as far if you hit when it is just floating in midair in front of you, as you would do if you hit it when it is pitched at a very high speed.

I am uncertain about what variables are of importance in this case. The velocity of the ball and the bat matters, however.

Homework Equations

I believe that the relevant equations in this problem could be the following:

Impulse F$$\Delta$$t = m$$\Delta$$v
where F = force, $$\Delta$$t = contact time between ball and bat, m = mass and $$\Delta$$v = change in velocity

I have also been thinking that maybe the coefficient of restitution e could be used in some way. Also, I have tried to find any information about elastic potential energy, and if this could be used to calculate the velocity after collision between bat and ball,

The Attempt at a Solution

I have been performing an experiment to find how the contact time between ball and bat varies when the ball is thrown at different speeds. This experiment was however slightly unsuccessful due to poor equipment. What I found was that the contact time was in general 0.0040 seconds for a pretty slow shot. This could be wrong, though. Otherwise, I seemed to find that the contact time decreased as the relative speed between ball and bat was increased during the collision. Could this be correct?

Anyway, one thing I've been wondering about is why the ball can be returned with higher speed when it approaches with higher speed. Is not more force needed to compensate for the bigger change in momentum (or velocity) of the ball? I can't seem to find any solution to this. Please someone, help me. I'd really like to be able to do this. Feel free to ask about further details in my investigation if it is of interest. Thanks on beforehand.

 

[Astronuc]

Otherwise, I seemed to find that the contact time decreased as the relative speed between ball and bat was increased during the collision. Could this be correct?

Yes. The speed of the bat will be a factor. Otherwise, for the same swing of the bat each time, the deformation of the tennis ball will increase as the speed of the ball increases, so more energy is stored in the ball.

One could measure the effect of the speed of the ball by dropping it vertically with different speeds and measuring the recoil/rebound.

 

[EriqueSherman]

All right. I was however surprised that the contact time didn't increase when the speed increased. I had expected that the more you deform the ball the longer is the contact time between ball and bat since the bat is "buried" deeper into the ball.

 

[EriqueSherman]

I've been thinking a little now, and also looking for more information on the internet. I found this formula from two different sources:

V$$_{f}$$ = eV$$_{ball}$$ + (e+1)V$$_{bat}$$

where
V$$_{f}$$ is the velocity of the ball after collision with bat
V$$_{ball}$$ is the velocity of the ball before the collision
V$$_{bat}$$ is the velocity of the bat before the collision
e is the coefficient of restitution

I found the following definition of the coefficient of restitution (from wikipedia):

This is for a collision between two moving objects. The velocities involved are those of the bat and the ball before and after the collision. The problem is I have no means of measuring the speed of the bat and the ball like this.
If you are however bouncing something against a rigid surface, like a floor, this could be used instead:

which is the object's velocity after impact divided by its velocity before impact.
So I assume that the collision between the bat and the ball does not bend the bat (because a tennis ball is relatively soft compared to for example a baseball), and hence the bat acts as a rigid surface. I have found e to be approximately 0.75 for a tennis ball (using law of energy conservation to find the two velocities when dropped from a particular height)

Is anyone familiar with this (this being the coefficient of restitution and this type of physics problem)? Does my method appear reasonable?