Welcome to Science Buddies. This is a very interesting project idea. Thank you for your question. I had to brush up on my knowledge of the Coefficient of Restitution (COR) parameter. I had a lot of fun researching it.
The physics of a ball vs. a bat depends not upon actual speeds of the ball and bat, but rather the ratio of the speeds of each. The numbers may change in a fast vs. slow collision, but the physics is the same. That is where COR (denoted by the symbol “e”) comes in. It is the ratio, a number between 0 and 1, of the relative speed of the bat and ball after collision over the relative speed before (e =Vout/Vin). Theoretically, a bat with a e = 0.50 and swung at a stationary ball at 5m/s and again at 10m/s would cause the ball to exit the collisions at 10m/s and 20m/s respectively. As I understand, you want to test and verify this experimentally. Additionally, you want to see if the COR of a bat changes over time with use.
The easiest way I can see to set up a test would be to mount the bat like a pendulum (i.e. vertically on a rotating hinge by the handle grip). Let gravity swing the bat for you. Place the ball on a tee so that the barrel of the bat makes contact when it is exactly at the bottom of the swing arch. By placing a marked scale behind the bat/ball setup and filming the collision, you will be able to determine velocity of the bat at the bottom of the swing, just prior to contact with the ball (Vin) and velocity of the bat and ball after collision (Vout). Remember, after collision, the velocity of the bat (Va) is reduced while the ball gains velocity (Vb). Vout is the difference (Vout = Vb-Va).
With this setup, you can place the ball at various heights, simulating a batter striking the ball in the “sweet spot” vs. close to the handle or close to the tip. Additionally, I think it would be much easier to obtain a consistent swing and hit vs. trying to drop a ball onto the bat and trying to hit the barrel dead center every time and in the same location length wise. If you want to test the results of a tipped or grounded ball (i.e. hitting the ball off center) then all you have to do is move your tee over a little bit. If you really wanted to be accurate and you have access to them, you could mount accelerometers and/or use a radar/laser gun on the system to determine velocities.
A few comments on your movable axis setup: I think it’s unnecessary. I see where you are thinking of simulating the break in the wrist of a batter that may change the results. Really all this “break” does is change the Vin of the system. If the bat recoils back, this just translates to a slower collision speed and probably an off-axis hit. All of which can be simulated and changed in the pendulum setup described above in a much more controlled manner. The “trampoline” effect of the bat (i.e. the natural resonance) will be the same regardless. Additionally, with the pendulum setup, you could mount the bat on the hinge at various points on the handle to simulate a batter who has “choked down” on the bat.
Now as for the math, you are performing a collision experiment. There is virtually no difference between a bat hitting a ball and a ball hitting a ball. You can use the same math in both cases. The difference is this: The mass of the baseball will remain constant. However, the mass of the bat will depend on where you hit the ball along the barrel. This is because you don’t hit with the entire bat, only the small part that comes in contact with the ball and the whole mass of the bat is not involved in the collision, unless it occurs exactly at the center of mass of the bat. A hit anywhere else along the barrel and the effective mass of the bat will be less than its whole mass. If you want to calculate the forces involved and/or the momentum then we refer to the familiar equation F=ma. For the bat vs. ball collision, the force applied by the bat is an average force over the time they are in contact with each other (delta_t). This is because, at the start of the collision, the initial force is small and it increases to some max value at the height of contact and then drops off again at the end of the collision and the ball leaves with new velocity and momentum vectors.
This is alot, so I hope it makes sense. Let us know if you have additional questions.
I hope this helps.
"As the circle of light increases, so does the circumference of darkness around it."
~ Albert Einstein