Question

The ratio of the radii of planets $A$ and $B$ is ${{K}_{1}}$ and the ratio of acceleration due to gravity on them is ${{K}_{2}}$. The ratio of escape velocities from them will be:
A.${{K}_{1}}{{K}_{2}}$
B.$\sqrt{{{K}_{1}}{{K}_{2}}}$
C.$\sqrt{\dfrac{{{K}_{1}}}{{{K}_{2}}}}$
D.$\sqrt{\dfrac{{{K}_{2}}}{{{K}_{1}}}}$

Answer 1

Hint: To solve this problem we need to know about the formula for escape velocity first. The formula for escape velocity is $v=\sqrt{2gR}$, here $g$ is the acceleration due to gravity on the planet and $R$ is the radius of that particular planet. By using this formula, we can find out the ratio of escape velocities from both the planets as mentioned in the question.
Formula used: $v=\sqrt{2gR}$

Complete answer:
Let us find out the escape velocity of planet $A$ that has a radius ${{R}_{A}}$ and the acceleration due to gravity on that planet is ${{g}_{A}}$. Thus, the escape velocity from planet $A$ will be:
${{v}_{A}}=\sqrt{2{{g}_{A}}{{R}_{A}}}$
Similarly, the escape velocity of planet $B$ that has a radius ${{R}_{B}}$ and the acceleration due to gravity on that planet is ${{g}_{B}}$ can be found out with the help of the same formula. Thus, the escape velocity from planet $B$ will be:
${{v}_{B}}=\sqrt{2{{g}_{B}}{{R}_{B}}}$
The ratio of escape velocities from both the planets will be as follows:
$\begin{align}
  & \dfrac{{{v}_{A}}}{{{v}_{B}}}=\dfrac{\sqrt{2{{g}_{A}}{{R}_{A}}}}{\sqrt{2{{g}_{B}}{{R}_{B}}}} \\
 & \Rightarrow \dfrac{{{v}_{A}}}{{{v}_{B}}}=\sqrt{\dfrac{{{g}_{A}}{{R}_{A}}}{{{g}_{B}}{{R}_{B}}}} \\
\end{align}$
It is given that the ratio of the radii of planets $A$ and $B$ is ${{K}_{1}}$ which means that $\dfrac{{{R}_{A}}}{{{R}_{B}}}={{K}_{1}}$ and ratio of acceleration due to gravity on them is ${{K}_{2}}$ which implies that $\dfrac{{{g}_{A}}}{{{g}_{B}}}={{K}_{1}}$. Hence:
$\begin{align}
  & \dfrac{{{v}_{A}}}{{{v}_{B}}}=\sqrt{\dfrac{{{g}_{A}}{{R}_{A}}}{{{g}_{B}}{{R}_{B}}}} \\
 & \therefore \dfrac{{{v}_{A}}}{{{v}_{B}}}=\sqrt{{{K}_{1}}{{K}_{2}}} \\
\end{align}$
The ratio of escape velocities from both the planets will be $\sqrt{{{K}_{1}}{{K}_{2}}}$.

Hence the correct option is $B$.

Note:
The escape velocity is the velocity with which if a particle is thrown then it would escape or defy the gravitational pull of the Earth. We can also say that the kinetic energy of the particle must be slightly greater than its gravitational potential energy in order to escape from the Earth.