This is one of those frustrating to "aha!" moments you'll experience in robotics. To understand the solution to the problem, it's best to approach this in a scientific manner.

#### Let's first take a look at the mechanics of the launcher:

1.) The gear drives the puncher back, using a linear slide. This process stretches the rubber band. Maximum tension is achieved at the tipping point, where the gear is cut. This is the moment of greatest potential energy. PE = 1/2 k x^2, where x is the distance between the two rubber band ends- in your case, the distance between the stand off on the puncher and the fixed standoffs on the five wide c-channel. k is your elasticity modifier- depending on the elasticity of the rubber band, you will store more or less energy.

2.) The gear "slips", converting the stored potential energy into kinetic energy proportional to the square of the remaining distance: KE = 1/2 k xfinal^2 - 1/2 k xremaining^2. This is very important! Because only kinetic energy is transferred to the ball, you want the puncher to hit the ball when KE is at it's peak. This occurs when xremaining = 0, as all PE has been converted to KE! Looking at your design, it's apparent that the the puncher is prevented from ever achieving maximum KE- which would be when all standoffs are in line with each other.

3.) Before the stopping mechanism can act, the puncher hits the ball, transferring energy to the ball. This means that the kinetic energy the stopping mechanism needs to absorb is equivalent to KEpuncher - KEball + PE remaining -PEcurrent. I'll break that down in a second.

#### So! Time to calculate a solution!

Lets look at that formula again: KEpuncher - KEball + PE remaining -PEcurrent = KE to be absorbed by stop:

KEpuncher = 1/2 k xfinal^2 - 1/2 k xremaining^2

KEball = energy transferred to ball.

How to calculate KEball? The puncher hitting the ball is similar to an inelastic collision. The total moment of the system must remain intact. Therefore, we calculate the total moment:

before the collision: Mpuncher*Vpuncher

and the total moment

after the collison: Mpuncher&ball * Vpuncher&ball

Moment is conserved:

moment before = moment after = > Mpuncher&ball*Vpuncher&ball = Mpuncher*Vpuncher

therefore: Vpuncher&ball = (Mpuncher*Vpuncher)/Mpuncher&ball, with Vpuncher&ball = Vball

to find Vpuncher:

Ideally, all PE is converted to KE (as this is best for shooting), so:

KEpuncher = PEmax = > 1/2 Mpuncher *Vpuncher^2 = 1/2 k * xfinal^2

Vpuncher = ((k*x^2)/M)^(1/2)

Finally, KEball:

1/2*Mball*(Vball^2)

PE Remaining - PEcurrent should be 0 in an efficient launcher (as all PE converted to KE), so we are left with:

KEabsorbed = KEpuncher&ball - KEball

#### Finishing up:

An astute observer will note that not all energy was transferred to the ball- only some of it. This is quite important, as you will now realize that the ball can never fully prevent damage to the mechanism.

So now we know how much energy needs to be absorbed. The only thing left to do is to figure out how to absorb the energy, or minimize the damage (work) it can do.

In order to do so, we redefine damage to something a bit more scientific: Work.

Work is force integrated over a spacial dimension. Ignoring the jargon, and reflecting our problem, we can simplify this to:

Work = force * distance = Mass * Acceleration * Distance = Mass * (Distance * (1/Time)) * Velocity

#### This tells us we can minimize Work/Damage by:

- 1.) Decreasing the
**Mass** - 2.) Decreasing the
**velocity** the puncher has when hitting the stopping mechanism - 3.) Increasing the
**time** the puncher takes before coming to a halt once it has hit the stopping mechanism

#### Option 1:

Self explanatory, but if we decrease Mass, we also decrease how effectively we launch the ball. So, not viable.

#### Option 2:

Viable, but we would need the velocity to only be limited after contact has been made with the ball. This is possible: place the stopping mechanism in such a way that the rubber bands begin to pull the puncher back, away from the ball (after the ball has been hit) before the puncher hits the stopping mechanism. This solution also incorporates option 3.

#### Option 3:

Increasing the time taken to stop the puncher decreases the force exerted on the stopping mechanism at any given time. Example: laying a hammer on a shelf never breaks the shelf, but throwing the hammer onto the shelf at mach 2 would put a hole through the shelf. A solution to the problem would be feathering the stopping mechanism with cushioning material, rubber bands that act against the opposing force, etc.

We are especially interested in force, as it is force that causes damage to your system. Therefore, limiting force of impact limits damage done.