This task involves analyzing a collision between two blocks, $m_1$ and $m_2$, with a spring (with spring constant $k$) attached to $m_2$. Here's how to approach the questions:

Analysis and Solutions

a) Is the collision elastic or not?
To determine if the collision is elastic:

1. Elastic collision implies both kinetic energy and momentum are conserved.
2. Inelastic collision involves momentum conservation, but not necessarily kinetic energy conservation.

Here, as the spring compresses, potential energy is stored in the spring and then released, implying no energy loss to heat or other factors. Thus, the collision is elastic, assuming an ideal spring and no energy loss.

b) Maximum compression of the spring ($x_{\text{max}}$):
The maximum compression occurs when the relative velocity of the blocks equals zero momentarily, meaning both blocks move together as one unit.

1. Initial kinetic energy of the system: $E_{\text{kin, initial}} = \frac{1}{2} m_1 v_1^2$

2. At maximum compression, all the kinetic energy of $m_1$ is converted into the spring's potential energy: $E_{\text{spring}} = \frac{1}{2} k x_{\text{max}}^2$

3. By conservation of energy: $\frac{1}{2} m_1 v_1^2 = \frac{1}{2} k x_{\text{max}}^2$

4. Solve for $x_{\text{max}}$: $x_{\text{max}} = \sqrt{\frac{m_1 v_1^2}{k}}$

c) Final velocities of the blocks ($v_1'$ and $v_2'$):
By conservation of momentum and the fact that the collision is elastic, we use these equations:

1. Momentum conservation: $m_1 v_1 = m_1 v_1' + m_2 v_2'$

2. Velocity relationship in elastic collisions: $v_1 - v_2 = -(v_1' - v_2')$

Combine these equations to solve for $v_1'$ and $v_2'$: $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1$ $v_2' = \frac{2 m_1}{m_1 + m_2} v_1$

Summary of Results

• a) The collision is elastic.
• b) Maximum compression of the spring: $x_{\text{max}} = \sqrt{\frac{m_1 v_1^2}{k}}$
• c) Final velocities of the blocks: $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1, \quad v_2' = \frac{2 m_1}{m_1 + m_2} v_1$

Let me know if you'd like further details!
Follow-up questions:

1. How does the spring constant $k$ affect $x_{\text{max}}$?
2. What happens if $m_1 = m_2$?
3. How is energy shared between the blocks after the collision?
4. Can this method handle a non-ideal spring?
5. How would damping forces affect the analysis?

Tip: Always verify whether energy losses occur when analyzing real-world systems!