M(kicker) * V(kicker) = M(ball) * V(ball)

That is close but not quite correct. Momentum is conserved, and energy is conserved.

Therefore:

**Conservation of Momentum**

M(kicker)*V(kicker[before]) = M(Ball)*V(Ball) + M(Kicker)*V’(kicker)

Where V(kicker[after]) is the speed of the kicker after hitting the ball (do not assume this is 0).

**Conservation of Energy**

KE(kicker[before]) = KE(ball) + KE(Kicker[after]) + Eint

(Where KE = 1/2mv^2 and Where Eint is the internal energy lost to deformation of the ball, I suppose you could include others like heat as well, but I don’t see too much heat being created here). I ran an experiment in my garage (bouncing a soccer ball on my cement floor, recording with a camera, and analyzing the height of each bounce) and saw that roughly 60% of the kinetic energy of the soccer ball was lost on each bounce. I have not run equations taking this loss into effect, but my guess is that Eint = .4*KE )

Collisions can be elastic, inelastic or somewhere in between. Elastic means that kinetic energy is completely conserved (Eint in this case would be 0). Inelastic means kinetic energy is not conserved.

For sake of equations, lets assume the collision between a kicker and a soccer ball is completely elastic, so Eint = 0. You can then solve the two above conservation equations (took me a full page) but you get in the end:

V(ball) = 2 * V(kicker) * M(kicker) / ( M(ball) + M(kicker) )

Lets take the case where M(kicker) is much greater than the mass of the ball. Taking a limit as M(kicker) -> inf, Then:

V(ball) = 2 * V(kicker)

This is the maximum velocity you can ever hope to achieve with the ball (twice that of the kicker). As Walter Lewin (MIT) would say, “very non-intuitive.” Nevertheless, don’t concentrate on making your kicker too massive. The real key is balancing having enough mass and speed, as mentioned before using F = ma where your kicking force will be constant (if you are using a spring/elastics and pull it back to the same spot each time).

Yet this doesn’t take into account the loss of energy due to Eint. You’d have to re-solve all the equations for this. Or, just plan on designing something about twice as powerful as it is in theory.

This is the point where I stopped doing equations (trying to calculate everything was getting quite ridiculous) and just built a kicker prototype, which actually worked quite well at throwing balls across the room and breaking ceiling tiles. We are going with a spring loaded pullback (our prototype used bungee cords) Still working out details for a fast and strong winch to pull it back. Our wish is to have the release be adjustable, this will give use the advantage of selecting how hard we want to hit it. If all else fails, we can use a (probably simpler) system to release it at the same spot every time.