The task involves analyzing a collision with a spring ("Stoß mit Feder"). I'll solve this step by step:

Problem Overview:

We have two blocks:

• Block 1: Mass $m_1$, moving with initial velocity $v_1$.
• Block 2: Mass $m_2$, initially at rest.
• A spring (with spring constant $k$) is attached to Block 2. The spring compresses as Block 1 collides with Block 2. The system is frictionless.

We need to:

1. Determine if the collision is elastic or not.
2. Calculate the maximum compression of the spring during the collision.
3. Compute the final velocities of both blocks after the collision.

Solution Steps:

a) Is the collision elastic or not?

The collision is elastic if:

• Kinetic energy is conserved.
• Momentum is conserved.

Since no energy is lost (e.g., to heat or deformation), and the system has a frictionless surface, the collision is elastic.

b) Maximum compression of the spring ($x_{\text{max}}$):

During maximum compression:

• Both blocks move with the same velocity $v_{\text{common}}$ (relative velocity is zero at this point).
• Use conservation of momentum to find $v_{\text{common}}$: $m_1 v_1 = (m_1 + m_2) v_{\text{common}}$ $v_{\text{common}} = \frac{m_1 v_1}{m_1 + m_2}$

Now, consider the kinetic energy lost during compression. This energy is converted into potential energy of the spring: $\Delta E = \frac{1}{2} m_1 v_1^2 - \frac{1}{2} (m_1 + m_2) v_{\text{common}}^2$ $\Delta E = \frac{1}{2} k x_{\text{max}}^2$

Substitute $v_{\text{common}}$ into $\Delta E$: $\frac{1}{2} m_1 v_1^2 - \frac{1}{2} (m_1 + m_2) \left(\frac{m_1 v_1}{m_1 + m_2}\right)^2 = \frac{1}{2} k x_{\text{max}}^2$

Solve for $x_{\text{max}}$: $x_{\text{max}} = \sqrt{\frac{m_1 m_2 v_1^2}{k (m_1 + m_2)}}$

c) Final velocities of the blocks ($v_1'$, $v_2'$):

Using the equations for an elastic collision:

1. $v_1'$ (final velocity of $m_1$): $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1$
2. $v_2'$ (final velocity of $m_2$): $v_2' = \frac{2 m_1}{m_1 + m_2} v_1$

Final Results:

1. The collision is elastic.
2. Maximum spring compression: $x_{\text{max}} = \sqrt{\frac{m_1 m_2 v_1^2}{k (m_1 + m_2)}}$
3. Final velocities of the blocks: $v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1$ $v_2' = \frac{2 m_1}{m_1 + m_2} v_1$

Would you like further details on any of these steps?

Related Questions:

1. How is energy conserved during a perfectly elastic collision?
2. How would adding friction affect the system's outcome?
3. What happens if the collision is perfectly inelastic instead of elastic?
4. Can the spring constant $k$ influence the final velocities of the blocks?
5. What role does the mass ratio $m_1/m_2$ play in the collision dynamics?

Tip:

In elastic collisions, the relative velocity of the two objects before and after the collision is equal in magnitude but opposite in direction!