Question

Two bodies of masses $\mathrm{m}$ and $4\mathrm{m}$ are moving with equal linear momenta. The ratio of their kinetic energies is

• A. $1:4$
• B. $4:1$
• C. $2:1$
• D. $1:2$

Answer 1

The correct option is B

$4:1$

Step 1. Given data

Mass of first body is given as $\mathrm{m}$, mass of second body is given as $4\mathrm{m}$ which are moving with equal linear momenta ${\mathrm{p}}_1$ and ${\mathrm{p}}_2$ as shown in above figure.

We have to find the ratio of kinetic energy.

Step 2. Formula to be used

Only moving objects can possess kinetic energy and momentum. If we adapt the mass time squared velocity definition, kinetic energy and momentum are related to one another and we will get,

$\mathrm{E}=\frac{{\mathrm{p}}^2}{2\mathrm{m}}$

Here, $\mathrm{p}$ is momentum, $\mathrm{m}$ is the mass and $\mathrm{E}$ is energy.

Step 3. Find the ratio of kinetic energy.

Let,

${\mathrm{p}}_1=\sqrt{2\mathrm{mK}.{\mathrm{E}}_1}$

${\mathrm{p}}_2=\sqrt{2\times 4\mathrm{mK}.{\mathrm{E}}_2}$

$=\sqrt{8\mathrm{mK}.{\mathrm{E}}_2}$

If ${\mathrm{p}}_1={\mathrm{p}}_2$ for two bodies, we get,

$\sqrt{2\mathrm{mK}.{\mathrm{E}}_1}=\sqrt{8\mathrm{mK}.{\mathrm{E}}_2}$

Squaring on both sides.

${\left(\sqrt{2\mathrm{mK}.{\mathrm{E}}_1}\right)}^2={\left(\sqrt{8\mathrm{mK}.{\mathrm{E}}_2}\right)}^2$

$2\mathrm{mK}.{\mathrm{E}}_1=8\mathrm{mK}.{\mathrm{E}}_2$

$\frac{2\mathrm{mK}.{\mathrm{E}}_1}{8\mathrm{mK}.{\mathrm{E}}_2}=1$

$\frac{1}{4}\frac{\mathrm{K}.{\mathrm{E}}_1}{\mathrm{K}.{\mathrm{E}}_2}=1$

$\frac{\mathrm{K}.{\mathrm{E}}_1}{\mathrm{K}.{\mathrm{E}}_2}=\frac{4}{1}$

Therefore, the ratio of kinetic energies are $\frac{4}{1}$.

Hence, option $\mathrm{B}$ is correct answer.