Question

Two particles of masses 1 g and 4 g have kinetic energies. What is the ratio between their momenta?

• A. 1:4
• B. 1:8
• C. 1:2
• D. _1:16

Answer 1

The correct option is C

1:2

Step 1: Given data

According to the question we have given that

Mass of a first body, $m_1=1g$

Mass of a second body, $m_2=4g$

The kinetic energy of both masses is the same, i.e $K.E_1=K.E_2=K.E$

Step 2: Relation between momentum and kinetic energy

Use the relation between linear momentum$\left(P\right)$ and kinetic energy$(K.E)$

$$\begin{gathered} K.E=\frac{P^2}{2m} \\ \Rightarrow P=\sqrt{2mK.E} \end{gathered}$$

Step 3: Finding the ratio of momentum

$$\begin{gathered} P=\sqrt{2mK.E} \\ \Rightarrow P_1=\sqrt{2m_{1}K.E} \\ \Rightarrow P_2=\sqrt{2m_{2}K.E} \\ Takingtheratioofmomentum \\ \Rightarrow \frac{P_1}{P_2}=\frac{\sqrt{2\times 1\times K.E}}{\sqrt{2\times 4\times K.E}} \\ \Rightarrow \frac{P_1}{P_2}=\sqrt{\frac{1}{4}} \\ \Rightarrow \frac{P_1}{P_2}=\frac{1}{2} \end{gathered}$$

We conclude that the ratio of the magnitudes of their momenta is $1:2$.

Hence, Option C is correct.