Question

A stone is dropped from a height $\text{h}$. It hits the ground with a certain momentum $p$. If the stone is dropped from a height $100\%$ more than the previous height, the momentum when it hits the ground will change by

• A. $68\%$
• B. $41\%$
• C. $200\%$
• D. $100\%$

Answer 1

The correct option is B

$41\%$

Step 1: Given data

$\text{momentum}=p$

$h_1=h$

$h_2=2h$

Step 2: Formula used

Velocity $v=\sqrt{2gh}$

Momentum $p=mv$

Here,

$h$ is the height.

$g$ is the gravitational acceleration.

$m$ is mass.

Step 2: Calculate the percentage change in momentum.

$\Rightarrow p=m\sqrt{2gh}$

$$\begin{gathered} \frac{p_2}{p_1}=\frac{\sqrt{h_2}}{\sqrt{h_1}} \\ \frac{p_2}{p_1}=\sqrt{\frac{2h}{h}} \\ {p}_2=\sqrt{2}{p}_1 \end{gathered}$$

Now,

$$\begin{gathered} \%\text{change}=\frac{p_2-p_1}{p_1}\times 100 \\ =\frac{\sqrt{2}p-p}{p}\times 100 \\ =41\% \end{gathered}$$

Hence, Option (B) is correct.