A bullet of mass m embeds itself in a block of mass M resting on a smooth horizontal surface, attached to a spring of force constant k. If the initial speed of the bullet is `v_(0)` along horizontal, find (a) the maximum compression of the spring and (b) the time for the bullet - block system to come to rest.

Explanation

To solve the problem step by step, we will break it down into two parts: finding the maximum compression of the spring and calculating the time for the bullet-block system to come to rest.

### Part (a): Maximum Compression of the Spring

1. **Conservation of Momentum**:
When the bullet of mass \( m \) embeds itself into the block of mass \( M \), we can use the conservation of momentum to find the velocity \( v \) of the bullet-block system immediately after the collision.
\[
\text{Initial momentum} = \text{Final momentum}
...