Animated Solution for Physics - Work, Energy, and Power: The block of mass $M$ moving on the frictionless horizontal surface collides with the spring of spring constant $k$ and compresses it by length $L$. The maximum momentum of the block after collision is

Imagine you are watching a block of mass $M$ sliding gracefully across a perfectly smooth, frictionless surface. It is heading straight for a spring with a spring constant $k$. When it hits, a beautiful transfer of energy begins. The block compresses the spring, slowing down until it momentarily stops. At this exact instant, the spring is compressed by a maximum length $L$.

But the story doesn't end there. The spring, now loaded with potential energy, pushes back, accelerating the block in the opposite direction. Our goal is to find the maximum momentum of the block after this collision.

To find the maximum momentum, we first need to understand when it occurs. Momentum is defined as $p=Mv$. Since the mass $M$ is constant, the momentum is at its absolute maximum when the velocity $v$ is at its maximum.

When does the block reach its maximum velocity? Right after the spring has fully expanded back to its natural length, pushing the block away. At this point, all the energy stored in the spring has been transferred back to the block as kinetic energy.

Because the surface is frictionless, we can confidently apply the Principle of Conservation of Mechanical Energy. The total energy of the system remains constant throughout the entire process.

At the moment of maximum compression, the block is momentarily at rest, so its kinetic energy is zero. All the energy is stored as elastic potential energy in the spring:

When the block leaves the spring, the spring is back to its natural length, so its potential energy is zero. All the energy is now kinetic energy of the block:

Equating the two, we get our master equation:

Now, we could solve for velocity $v$ and then multiply by mass $M$ to find the momentum. But there is a much more elegant trick! We can express kinetic energy directly in terms of momentum using the relation:

Let's substitute this directly into our energy conservation equation. This small substitution saves us from messy algebra and takes us straight to the answer:

Now, the path is clear. We just need to isolate $p$. First, let's cancel the $\frac{1}{2}$ from both sides:

Next, multiply both sides by $M$ to get $p^2$ by itself:

Finally, take the square root of both sides to find the maximum momentum:

And there we have it! The maximum momentum of the block is $L\sqrt{Mk}$, which perfectly matches option (a).

Always remember, expressing kinetic energy as $\frac{p^2}{2M}$ is a powerful tool in your physics arsenal, especially in collision and energy problems!