Animated Solution for Physics - Work, Energy, and Power: A boy is rolling a $0.5\text{ kg}$ ball on the frictionless floor with the speed of $20{\text{ ms}}^{-1}$. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now ?

A Rolling Ball and an Obstacle

Imagine you are standing in a perfectly smooth, frictionless hallway. You roll a $0.5\text{ kg}$ ball across the floor, and it zips away at a brisk speed of $20{\text{ ms}}^{-1}$. Suddenly, it strikes an obstacle and deflects off its original path.

But this wasn't a perfect bounce. The problem tells us that after the deflection, the ball retains only $5\%$ of its initial kinetic energy. Our mission is to find out exactly how fast the ball is moving after this highly inelastic collision.

To understand what happens after the crash, we first need to know how much energy the ball carried into it. The energy of motion is called Kinetic Energy, and it is governed by the classic equation:

Let's plug in the initial conditions. We know the mass $m=0.5\text{ kg}$ and the initial speed $u=20{\text{ ms}}^{-1}$. Substituting these values gives us:

Squaring the speed gives us $400$. Multiplying that by $0.5$ gives $200$, and taking half of that leaves us with exactly $100\text{ J}$. So, the ball approached the obstacle carrying a solid $100\text{ Joules}$ of kinetic energy.

Where Did the Energy Go?

When the ball hits the obstacle, it undergoes a severe energy loss. The problem states that only $5\%$ of the initial kinetic energy survives the impact.

Calculating $5\%$ of $100\text{ J}$ is straightforward:

In a fraction of a second, $95\text{ Joules}$ of energy vanished from the ball's motion! Where did it go? In the real world, this energy is converted into the loud thud of the impact, the microscopic deformation of the ball and the obstacle, and a slight increase in thermal energy (heat).

Now that we know the ball is limping away with only $5\text{ Joules}$ of kinetic energy, we can work backward to find its new speed, let's call it $v$. We set up our kinetic energy equation once more:

Substitute the mass back in:

To isolate $v^2$, we divide both sides by $0.25$ (which is the same as multiplying by $4$):

Finally, we take the square root of $20$. We know that $\sqrt{16}=4$ and $\sqrt{25}=5$, so our answer must lie right in the middle.

Even though the ball lost a massive $95\%$ of its energy, its speed didn't drop by $95\%$. It dropped from $20{\text{ ms}}^{-1}$ to about $4.47{\text{ ms}}^{-1}$. This beautifully illustrates the non-linear relationship between speed and kinetic energy—because speed is squared in the formula, a small amount of speed still packs a proportional punch of energy!