Question

Two bodies with kinetic energies in the ratio $4:1$ are moving with equal linear momentum. The ratio of their masses is?

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

Answer 1

The correct option is C

$4:1$

Step 1: Given data

The ratio of two bodies with kinetic energies is = $4:1$

We have to find the ratio of their mass.

Step 2: Formula to be used

The energy of a body due to its motion is known as kinetic energy.

The kinetic energy is,

$\mathrm{K}=\frac{\mathrm{m}{\mathrm{v}}^2}{2}$

Here $\mathrm{m}$ is the mass of the body and $\mathrm{v}$ is velocity.

We know that, if a body or object is at rest condition, then its velocity will be zero and since, zero kinetic energy.

The relation of momentum of body is,

$\mathrm{p}=\mathrm{mv}$

Where p is momentum, m is mass and v is the velocity.

Step 3: Finding the ratio of masses
The relation between the momentum and kinetic energy is given by

$\mathrm{K}=\frac{{\mathrm{p}}^2}{2\mathrm{m}}$
From the given,

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

and also the linear momentum is the same for both bodies.

So,

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

$\frac{{\mathrm{K}}_1}{{\mathrm{K}}_2}=\frac{{{\mathrm{p}}_1}^2\times 2{\mathrm{m}}_2}{2{\mathrm{m}}_1\times {\mathrm{p}}_2^2}$

$\frac{{\mathrm{p}}_1^2\times 2{\mathrm{m}}_2}{2{\mathrm{m}}_1\times {\mathrm{p}}_2^2}=\frac{4}{1}$

$\frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}=\frac{4}{1}$

The ratio of their masses is = 4:1

Hence, option (C) is the correct answer.