Is Momentum Conserved When a Spring Unloads Between Two Masses?

[kyrillos]

Let's say there are two masses, attached together by a string, and there's a compressed spring in between them. When the string in between is cut off, the spring unloads, pushing both masses in opposite directions.

My thinking:
1. Their momentums will be equal to each other.
Their momentum before the string is cut is equal to zero (because they were not moving). So the sum of their final momentums should be also zero.
2. Their kinetic energies will be equal to each other.
The spring unloads in both direction at equal rates, so I assume that their kinetic energies must also be equal. (I couldn't find a mathematical evidence in my Giancoli book to see if I am actually right)

Are my conclusions right?

 

[Dale]

1. Their momentums will be equal to each other.
Their momentum before the string is cut is equal to zero (because they were not moving). So the sum of their final momentums should be also zero.

Yes, that is correct.

2. Their kinetic energies will be equal to each other.
The spring unloads in both direction at equal rates, so I assume that their kinetic energies must also be equal

This is only correct if the masses are equal. Can you think what will happen if the masses are not equal?

 

[kyrillos]

Yes, that is correct.

This is only correct if the masses are equal. Can you think what will happen if the masses are not equal?

Well:
0 = m₁v₁ + m₂v₂
v₁ = -m₂v₂/m₁

And:
0.5m₁v₁² + 0.5m₂v₂² = 0.5kx²

I don't know what to do next, since I also don't know the value of the elastic potential energy. Could you help me?

 

[Dale]

I don't know what to do next, since I also don't know the value of the elastic potential energy. Could you help me?

You can just leave the elastic potential energy as you have written it. So the next step is just to substitute

v₁ = -m₂v₂/m₁

into

0.5m₁v₁² + 0.5m₂v₂² = 0.5kx²

and then solve for the velocity.