Question

Two particles have masses of $m$ and $4m$ and their kinetic energies are in the ratio of $2:1$. What is the ratio of their linear momenta?

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{4}$
• D. None of these

Answer 1

The correct option is B

$\frac{1}{\sqrt{2}}$

Step 1: Given

1. The mass of two particles are $m$ and $4m$
2. The ratio of kinetic energies are $2:1$

Step 2: Solution

From the relation of kinetic energy and momentum

$KE=\frac{P^2}{2m}$

$KE=$The kinetic energy of the particle

$P=$ The momentum of the particle

$m=$ The mass of the particle

The kinetic energy of the first particle

$K{E}_1=\frac{{P_1}^2}{2m}$

The kinetic energy of the second particle

$K{E}_2=\frac{{P_2}^2}{8m}$

The ratio of the kinetic energy of the two particles

$\frac{K{E}_1}{K{E}_2}=\frac{2}{1}\left[Given\right]$

$$\begin{gathered} \Rightarrow \frac{\frac{{P_1}^2}{2m}}{\frac{{P_2}^2}{8m}}=\frac{2}{1} \\ \Rightarrow \frac{{P_1}^2}{{P_2}^2}=\frac{1}{2} \\ \Rightarrow \frac{P_1}{P_2}=\frac{1}{\sqrt{2}} \end{gathered}$$

Hence, the correct option is $B$.