Calculate Final Speeds Two Mass Spring System

[PCB]

I am trying to calculate the final speeds of two masses on either side of a compressed spring, when the spring is released (in a frictionless environment). The problem has similarities to a perfectly elastic collision, in that the potential energy in the compressed spring would be the result of the v and m of the two masses which compressed the spring. Any suggestions?

 

[bigfooted]

A spring with a mass is a harmonic oscillator, which is governed by a second order ODE for the displacement. Your case is simply two harmonic oscillators connected using the same spring.

Here is an example that explains your case.
http://vergil.chemistry.gatech.edu/notes/ho/node2.html
You can get the equations either by applying Newton's second law or conservation of energy.

 

[Doc Al]

Any suggestions?

What's conserved?

You can think of this as the reverse of a perfectly elastic collision--an explosion.

 

[PCB]

Thanks for the replies. 1. Unless I am wrong, the harmonic oscillator math will just give the frequency of oscillations, but not the speed of the masses (further assume the masses are not attached to the springs, so no oscillations occur). 2. Total energy of the system is conserved, of course. The energy that went into compressing the spring will equal the energy of the moving masses when the spring is released. 3. As you can see from my orginal post, I am thinking of this situation as a PEC, more specifically, the post collision part of the PEC

 

[Doc Al]

2. Total energy of the system is conserved, of course. The energy that went into compressing the spring will equal the energy of the moving masses when the spring is released.

Good. What else is conserved?

 

[PCB]

Ok, I play the game. Energy and momentum are conserved

 

[bigfooted]

The harmonic oscillator math will give you x(t) of the masses. Once you have that, you can calculate the velocity by derivation. The governing equations are the same for the fixed and the free mass problem up to the point where you reach the maximum spring displacement.

I also just realized that when you only need the velocities at the moment the masses detach from the springs, using the conservation equations is much easier - no need to solve the ODE's.

EDIT: so yes, just solve the energy and momentum equations as Doc Al suggested

 

[Doc Al]

Ok, I play the game. Energy and momentum are conserved

That's all you need.

 

[PCB]

Good advice from both of you, thank you. The trouble I am having now is that the oscillator math assumes the spring/mass is acting against a fixed wall. I forgot to specify that my masses are not equal in, er, mass.

 

[Doc Al]

Forget about the oscillator stuff--not relevant. Treat it like an explosion.