Animated Solution for Physics - Work, Energy, and Power: A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes a uncompressed spring and compresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is 10000 N/m. The spring compresses by

Imagine you are standing in a physics lab, watching a $2\text{ kg}$ block sliding across a rough horizontal floor. It's moving at a brisk $4\text{ m/s}$, heading straight for an uncompressed spring attached to a wall.

As soon as the block makes contact, a fierce battle of energy begins. The block wants to keep moving, but two formidable opponents are trying to stop it: the rough floor, which drags on the block with a kinetic friction of $15\text{ N}$, and the spring, which pushes back harder and harder as it gets compressed.

Eventually, the block is brought to a complete halt. The question is: how far did it manage to compress the spring before losing all its energy?

To solve this, we don't need complex kinematics or changing accelerations. We have a much more elegant tool: the Work-Energy Theorem.

The theorem tells us that the total change in kinetic energy is equal to the net work done on the object. Since the block comes to a complete stop, its final kinetic energy is zero. Therefore, its entire initial kinetic energy is "spent" doing work against the non-conservative friction force and storing elastic potential energy in the spring.

We can write this energy budget as:

Here, $x$ is the maximum compression of the spring. The left side is our total energy budget, and the right side shows exactly where that energy went.

Now, let's carefully substitute our known values into the master equation. We know the mass $m=2\text{ kg}$, the initial velocity $v=4\text{ m/s}$, the friction force $f_k=15\text{ N}$, and the spring constant $k=10000\text{ N/m}$.

Simplifying the left side, the half and the two cancel out, leaving us with $4^2$, which is $16$. On the right side, half of $10000$ is $5000$.

This is a classic quadratic equation. Let's rearrange it into the standard form $a{x}^2+bx+c=0$:

I know this quadratic equation looks a bit terrifying with that massive $5000$ coefficient, but let's take a breath and use the quadratic formula.

Calculating the discriminant: $15^2=225$, and $-4(5000)(-16)=+320000$. Adding them gives $320225$. The square root of $320225$ is approximately $565.88$.

Since a negative compression doesn't make physical sense here, we take the positive root:

We have our answer in meters! But wait, look at the options. They are all in centimeters. To convert, we simply multiply by $100$:

The block manages to compress the stiff spring by exactly $5.5\text{ cm}$ before coming to a dead stop. The elegance of energy conservation strikes again!