Question

If the kinetic energy of a body becomes four times its original value, then the new momentum will be more than its initial momentum by,

• A. 50%
• B. 100%
• C. 125%
• D. 150%

Answer 1

The correct option is B

100%

Step 1: Expression of kinetic energy

Kinetic energy is calculated using the expression, $K=\frac{1}{2}m{v}^2$

Kinetic energy is related to momentum as follows:

$K=\frac{P^2}{2m}$; where $P$=momentum and $m$=mass.

Step 2: Finding the change in momentum corresponding to the change in kinetic energy

Let K be the initial kinetic energy.

Given kinetic energy is increased four times.

So, new kinetic energy $K\text{'}=4K\to \left(1\right)$

In terms of new momentum $P\text{'}$ , we can write $K\text{'}=\frac{P{\text{'}}^2}{2m}$

Equation (1) can be written in terms of momentum as

$\frac{P{\text{'}}^2}{2m}=4\times \frac{P^2}{2m}$

$P{\text{'}}^2=4P^2$

Taking square root on both sides,

$P\text{'}=2P$

% change in $P$= $\frac{P\text{'}-P}{P}\times 100\%$

% change in $P$

$\begin{array}{rcl}& =& \frac{2P-P}{P}\times 100\%\\ & =& \frac{P}{P}\times 100\%\\ & =& 100\%\end{array}$

Thus, the new momentum will be 100% more than the initial momentum.

Hence, option (b) is correct.