Spring oscillation question HARD

[WINSTEW]

Homework Statement

Two objects are placed in a 1 dimensional box with springs of different constants at each end of the box. The box is made of wood and has a length of L. The first object has a mass of M and is initially compressed a distance of xinitial onto spring 1 of spring constant k. When M is released it collides elastically with a second object of 6M with is initially at rest. Both objects recoil in opposite directions and ignoring friction how far will the initial spring recompress when M strikes it. How far will the second spring compress when 6M hits it knowing that the spring constant is 2k. The total length of each spring is 1.5m at equilibrium.

Homework Equations

1/2kx^2
1/2mv^2
energy balance
momentum balance

The Attempt at a Solution

This is a problem I am having trouble getting started. I know the concept but I can't get it to the paper. The 1st spring has a potential energy that will be converted into a kinetic energy of the object M. This will give M momentum equal to its velocity X mass. This will collide with 6M transferring some energy. Since the momentum is conserved the the 6M will have a positive velocity and the m will now have a negative velocity and less momentum. This velocity can then be used to find out how much spring potential energy is needed to obtain this, and then find the compression needed.

 

[LowlyPion]

Homework Statement

Two objects are placed in a 1 dimensional box with springs of different constants at each end of the box. The box is made of wood and has a length of L. The first object has a mass of M and is initially compressed a distance of xinitial onto spring 1 of spring constant k. When M is released it collides elastically with a second object of 6M with is initially at rest. Both objects recoil in opposite directions and ignoring friction how far will the initial spring recompress when M strikes it. How far will the second spring compress when 6M hits it knowing that the spring constant is 2k. The total length of each spring is 1.5m at equilibrium.

Homework Equations

1/2kx^2
1/2mv^2
energy balance
momentum balance

The Attempt at a Solution

This is a problem I am having trouble getting started. I know the concept but I can't get it to the paper. The 1st spring has a potential energy that will be converted into a kinetic energy of the object M. This will give M momentum equal to its velocity X mass. This will collide with 6M transferring some energy. Since the momentum is conserved the the 6M will have a positive velocity and the m will now have a negative velocity and less momentum. This velocity can then be used to find out how much spring potential energy is needed to obtain this, and then find the compression needed.

Looks like you have the sense of the solution.

Since you know V_i in terms of the givens
1/2*k*X_i² = 1/2*M*V_i²

And you can figure the resulting velocities in terms of V_i, you should be able to work the effect of the resulting v1 back to spring1 and v2 to spring2. Looks like some things will cancel out.