Equation relating launch velocity to distance pulled back.

[sharky939]

Homework Statement

I've been tasked with designing a projectile launcher that must be built using either elastic bands or springs. The objective is to measure the launch velocity against the distance the projectile is pulled back before release. The launcher itself isn't supposed to be very large, as the projectile is quite small.
Basically, I need help in designing a very simple launcher, but my main need of help is discovering the equation that links launch velocity to the distance the elastic band/spring is pulled back.

Homework Equations

The Attempt at a Solution

My best attempt so far at the equation starts with measuring the force I aplly to pull back the projectile a certain distance

Then work out acceleration = force/mass (But I don't know what mass to use!)
Which then leads to final velocity (0) = Launch velocity (squared) + 2*acceleration*distance pulled back.

Any help would be greatly appreciated

 

[sharky939]

Right, I managed to make Force equal to Hooke's Law and the kinetic energy equation, KE = 1/2mv^2.
If I get the answer in Joules from this equation, which is equal to Newton metres, and I divide through by the distance pulled back, surely then the I'd get a value in Newtons, which would be a force.
That would then mean

F = kx = 1/2MV^2
Are there any holes in this method I've fallen into?? My value of V is the lauch velocity, worked out by using the horizontal component of the projectile motion of the projectile.

 

[mgb_phys]

Energy is force * distance so the energy stored in the lanucher is average force (hookes law) * distance pulled back, this will equal the KE of the object at launch, which to a first approximation is proprtional to the distance.