Help! Egg Launcher Project for High Schooler

[gogetagritz]

Not sure if this is the right place to post this, but will give it a shot.

I am helping out my little sis this weekend as she has an egg launcher project for high school. There are weight and dimension limitations of course, and I'm not quite clear on other rules (when I took this class ~8 years ago we made an air cannon, but I'm pretty sure those aren't allowed no). So I was thinking a slingshot.

I want to figure out the initial velocity after gravity was the only thing acting on it so I could find:
x(final)=V(intial)^2*sin(2theta)/g
and get an idea what kind of springy material I would need.

With acceleration=k (L2-L1)/m I was going around in circle trying figure out a way to get L2 (L2 being the extended length and L1 its rest legth) as a function of time, so I could integrate it. So I was thinking I could say this is a SHO and it has a solution of
x(t) = A cos (w0*t) ignoring phase because its staring at its maximum displacement? Then assuming that the prjectile have left the slingshot apparatus when L2=L1 or pi/2 I think, and just take v(intial) at pi/2 (ignoring time really knowing v is going to be at max) as v=A*w0. Giving me an initial veloicty as it exits the slingshot of:
v=(L2-L1)*(k/m)^(1/2)

Hope this is clear enough, I just wanted someone elses input. Or maybe someone knows a good way to get force(t) of a spring as it is contracting. The max dimensions are rectangular not square so I should probably figure out the difference between a larger k, smaller angle, and a 45 degree angle and a smaller k.

 

[andrevdh]

Measure the extensions of the slingshot as a result of adding loads to it (hanging vertically). This will give you a force vs extension graph. The area under the graph gives you the work done by the slingshot as it contracts (count the squares under the graph). Via the work-kinetic energy theorem you have that the change in kinetic energy of the projectile is given by the work done by the force exerted on it (ignoring a change in gravitational potential energy).

 

[gogetagritz]

Oh that's a good idea, i can just take potential to kenetic energy to momentum. I was planning on putting weights on it to see how big the nonlinear terms of the elastic constants are, but this will kill two birds with one stone. I suppose I could have just done a numerical integration from the begining.

 

[andrevdh]

Another way of doing it is kinematically (walking to and fro) - shoot the projectile horizontally and measure the range. Use projectile theory to calculate the launching speed. Maybe plot it as a function of the final stretch length? I am not clear on what you want to quantify - it gives me a headache to try and understand what you are saying in your original post.