Two bodies A and B of masses 4 kg and 16 kg respectively have the same kinetic energy. The ratio of the velocity of A and B is

The correct option is B

2:1

Given data:

Mass of the body A is ${\mathrm{m}}_{\mathrm{A}}=4\mathrm{kg}$

Mass of the body A is ${\mathrm{m}}_{\mathrm{B}}=16\mathrm{kg}$

Finding the ratio of the velocity of the body A and B

1. Kinetic energy is the energy that is possessed by the body when the body is in motion.

2. Mathematically we can see that,

$\mathrm{Kinetic}\mathrm{energy}=\frac{1}{2}\mathrm{x}\mathrm{mass}\mathrm{x}{\left(\mathrm{velocity}\right)}^2$

${\left(\mathrm{Velocity}\right)}^2=\frac{2\mathrm{x}\mathrm{Kinetic}\mathrm{energy}}{\mathrm{mass}}$

$\mathrm{Velocity}=\sqrt{\frac{2\mathrm{x}\mathrm{Kinetic}\mathrm{energy}}{\mathrm{mass}}}$

$\mathrm{V}=\sqrt{\frac{2\mathrm{x}\mathrm{K}.\mathrm{E}}{\mathrm{m}}}$

3. For body A,

${\mathrm{V}}_{\mathrm{A}}=\sqrt{\frac{2\mathrm{x}\mathrm{K}.{\mathrm{E}}_{\mathrm{A}}}{{\mathrm{m}}_{\mathrm{A}}}}$………(1)

4. For body B,

${\mathrm{V}}_{\mathrm{B}}=\sqrt{\frac{2\mathrm{x}\mathrm{K}.{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{m}}_{\mathrm{B}}}}$………(2)

5. Dividing the equations (1) by (2) we get,

$\frac{{\mathrm{V}}_{\mathrm{A}}}{{\mathrm{V}}_{\mathrm{B}}}=\frac{\sqrt{\frac{2\mathrm{x}\mathrm{K}.{\mathrm{E}}_{\mathrm{A}}}{{\mathrm{m}}_{\mathrm{A}}}}}{\sqrt{\frac{2\mathrm{x}\mathrm{K}.{\mathrm{E}}_{\mathrm{B}}}{{\mathrm{m}}_{\mathrm{B}}}}}$

6. Since kinetic energy o body, A and B are given equal i.e K.E_A = K.E_B = K.E

Where V_A and V_B are the velocities of body A and B,

$\frac{{\mathrm{V}}_{\mathrm{A}}}{{\mathrm{V}}_{\mathrm{B}}}=\frac{\sqrt{\frac{2\mathrm{x}\mathrm{K}.\mathrm{E}}{4\mathrm{kg}}}}{\sqrt{\frac{2\mathrm{x}\mathrm{K}.\mathrm{E}}{16\mathrm{kg}}}}=\sqrt{\frac{16\mathrm{kg}}{4\mathrm{kg}}}=\sqrt{\frac{4}{1}}=\frac{2}{1}$

Thus, the ratio of the velocity of A and B is 2:1.

Hence, option (B) is correct.