Difficult Elastic Collision problem between two springs

[physicsguy1011]

Homework Statement

HI, I’m trying to get better at physics problems while I’m out of school. I’ve been trying these problems over and over and cant seem to get them. I dont have a teacher to help me so I’m wondering if any of you could help. I was hoping I could see how you solve the problem, and what answers you get. Thanks for the help in advance!!!

Relevant Equations

Elastic potential energy
Linear momentum
Impulse
Conservation of momentum
Elastic collision

 

[jbriggs444]

That is not how the homework forum operates around here. You have to make your attempt first. Then we can offer assistance.

 

[physicsguy1011]

That is not how the homework forum operates around here. You have to make your attempt first. Then we can offer assistance.

This is where I am so far. I know that the potential energy of a spring is equal to the change in kinetic energy of an object that hits the spring, but that's for a non elastic object that comes into contact with the spring. I am lost when it comes to the fact that there are two elastic objects coming into contact with each-other. So my solution is not that in depth yet.

 

[jbriggs444]

Can you write down an equation that relates the compression of object 1 to the compression of object 2? Say, for instance the compression at a time when the force between the two objects is equal to F (any F will do).

Behind the scenes, I have realized that there are too many variables here and not enough equations. We know how much kinetic energy has to be absorbed, but we have two places it can be stored. We need to be able to write down another equation so we can figure out how it will end up being divided.

 

[kuruman]

Behind the scenes, I have realized that there are too many variables here and not enough equations. We know how much kinetic energy has to be absorbed, but we have two places it can be stored. We need to be able to write down another equation so we can figure out how it will end up being divided.

Doesn't Newton's 3rd law provide that equation? Note that $V_{cm}=0$.

 

[jbriggs444]

Doesn't Newton's 3rd law provide that equation? Note that $V_{cm}=0$.

Right. I was skipping past the invocation of Newton's third on the [perhaps mistaken] idea that it was too obvious to mention.

 

[kuruman]

The question is well-crafted in that some of the given information may be redundant, but it is internally consistent. One can get the relation between $x_1$ and $x_2$ either by using Newton's 3rd using the spring constants or by considering that the ratio of distances to the center of mass is and remains constant. Once the velocities are chosen, the spring constant ratio is fixed.

Now that I think about it, the two springs can be replaced with an equivalent spring in which case one does not need to worry about a second equation.

 

[physicsguy1011]

I’m not sure I understand this

 

[Chestermiller]

This is where I am so far. I know that the potential energy of a spring is equal to the change in kinetic energy of an object that hits the spring, but that's for a non elastic object that comes into contact with the spring. I am lost when it comes to the fact that there are two elastic objects coming into contact with each-other. So my solution is not that in depth yet.

They expect to to model this as two point masses with springs attached at their leading ends, with the unextended lengths of the springs equal to the radius of each ball.

 

[haruspex]

I am lost when it comes to the fact that there are two elastic objects coming into contact with each-other.

Your doubts are well founded. In reality, this is a problem in "massive springs", i.e. springs whose masses cannot be neglected. Worse, they are non-uniform.
This leads to all sorts of complexities, such as internal oscillations, varying area of contact and non-uniform compression.

To make it tractable, you are expected to model it as two point masses, each with a spring attached. You can do this either by considering the compression of each and the relationship between the two compression forces at any instant, or by using the formula for the effective spring constant for springs in series and treating them as a single spring.