Physics of Spring with Electrostatics

[SpringPhysics]

Homework Statement

Consider a simple 3-body problem in one dimension. One body (the projectile) has mass m. The second body (the target) is made up of two "atoms" each of mass M/2 (total mass = M). Assume an internal spring holds the two atoms together. If the projectile is sent toward the target and a second force makes the target reflect the projectile, what is the subsequent motion of the three bodies?

Homework Equations

F (spring) = -kx
F (electrical) = kQq/r^2

The Attempt at a Solution

The "second force" is an electrical force, i.e. the three particles are similarly charged. To simplify things, I will consider a lengthless and massless spring and ignore frictional forces. The question also implies that energy is not conserved between the projectile and the target (but the target system itself conserves energy).

I was thinking that as the projectile approaches the target with an initial speed (by Newton's First Law, speed should be constant because there are no external forces acting on it), the electrical force between the target and the projectile strengthens (due to an inverse relationship between electrical force, F, and distance, r, [F = kQQ/r^2]). As this occurs, the target is repelled by the projectile (from rest to an initial speed) and causes the nearest atom to compress the spring (because the projectile's electrical force on the farthest atom is weaker), causing the spring to gain potential energy [F = -kx]. Within the target, the potential energy of the spring will be converted into kinetic energy by bouncing back and forth with the two attached atoms, acting with and against the electrical force between the two atoms. Depending on the speed of the projectile, the distance the target travels before the projectile is repelled will differ (because the target is free to move).

Does the spring take the kinetic energy of the target or does the target continue to move indefinitely in space (by Newton's First Law)?
Does the spring eventually reach equilibrium due to the balance of electrical forces between the two attached atoms (which are similarly charged)?

Thanks, any help will be appreciated!

 

[Redbelly98]

Welcome to Physics Forums.

This looks like a conservation of energy and momentum problem. It would seem the projectile imparts some of its kinetic energy to one of the target masses initially.

Can you calculate the velocities of the projectile and target mass right after this collision? Express the answer in terms of m, M, and the projectile's initial velocity v₀.

 

[SpringPhysics]

The question asks more for a qualitative solution to the problem (i.e. determining the physics behind the problem and using physics logic to determine a general approximation of the motion of the system).

I originally thought that the "collision" conserves kinetic energy, but would it also be possible that the transferred kinetic energy from the projectile is held within the spring? If this is the case, wouldn't the mass eventually slow down because it loses kinetic energy? Or would the mass continue to move at a slower speed?

Sorry for the questions, I'm just a little lost about the physics behind the interactions. All I know is that a spring can convert the kinetic energy into potential energy, but would that potential energy be released back into kinetic energy to move the attached masses?

What I'm trying to say is:

O||||||O <-- O

<- O||||||O <-- O

<-- O|||||O O
(the point where the electrical force of the spring system repels the projectile; the spring is compressed)

<-- O|||O O -->

...and then the spring system either continues to move leftwards with a constant velocity while the spring undergoes harmonic motion or the spring system slows down while the spring's harmonic motion slows down.