Animated Solution for Physics - Work, Energy, and Power: If the kinetic energy of a moving body becomes four times of its initial kinetic energy, then the percentage change in its momentum will be

Energy and Momentum

Imagine a block of mass $m$ gliding smoothly across a frictionless surface with a velocity $v$. This simple motion carries two fundamental physical quantities: momentum ($p$) and kinetic energy ($K$).

Momentum is the measure of the "quantity of motion," defined simply as $p=mv$. Kinetic energy, on the other hand, is the energy of that motion, given by $K=\frac{1}{2}m{v}^2$.

To solve problems efficiently, we need a direct bridge between these two quantities. By substituting $v=\frac{p}{m}$ into the kinetic energy equation, we get the master relation:

This equation is incredibly powerful. It tells us that for a body of constant mass, the kinetic energy is directly proportional to the square of its momentum ($K\propto p^2$). Conversely, the momentum is directly proportional to the square root of the kinetic energy ($p\propto \sqrt{K}$).

The problem presents a fascinating scenario: the kinetic energy of the body becomes four times its initial value. Let's write this mathematically:

Since we know that $p\propto \sqrt{K}$, we can set up a ratio to find the new momentum:

Now, we substitute our given condition into the ratio:

The $K_1$ terms cancel out beautifully, leaving us with:

This means the final momentum is exactly double the initial momentum ($p_2=2p_1$).

We are asked to find the percentage change in momentum. The standard formula for percentage change is:

Let's plug in our new momentum $p_2=2p_1$:

The momentum has increased by exactly 100%. This makes perfect physical sense: doubling the momentum ($2\times$) results in quadrupling the kinetic energy ($2^2=4\times$).