Question

There are two planets. The ratio of the radius of the two planets is $K$ but the ratio of acceleration due to gravity of both planets is $g$. What will be the ratio of their escape velocity?

• A. ${\left(kg\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$
• B. ${\left(kg\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$
• C. ${\left(kg\right)}^2$
• D. ${\left(kg\right)}^{-2}$

Answer 1

The correct option is A

${\left(kg\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Step 1: Given data

The ratio of the radius of two planet$=K$

The ratio of acceleration$=g$

Step 2: Formula used

The escape velocity of the planet can be computed using the following formula:

$V_e=\sqrt{2gR},$ where $V_e=$escape velocity, $g=$gravitational constant, and $R=$radius of planet

Step 3: Compute the ratio of escape velocity

Suppose the ratio of two planets is $R_1,R_2$ and the accelerations because of gravity are $g_1,g_2$ respectively.

So, the escape velocity of the first planet $V_{e1}=\sqrt{2g_1{R}_1}\dots \dots \dots \dots \dots \left(1\right)$

And the escape velocity of the second planet $V_{e2}=\sqrt{2g_2{R}_2}\dots \dots \dots \dots \dots \left(2\right)$

So, the ratio of escape velocity is, that is from $\left(1\right)\mathrm{and}\left(2\right)$,

$\frac{V_{e1}}{V_{e2}}=\sqrt{\frac{2g_1{R}_1}{2g_2{R}_2}}$

It is understood that $\frac{R_1}{R_2}=K\mathrm{and}\frac{g_1}{g_2}=g$. So,

$$\begin{gathered} \frac{V_{e1}}{V_{e2}}=\sqrt{Kg} \\ \frac{V_{e1}}{V_{e2}}={\left(Kg\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \end{gathered}$$

Thus, the ratio of their escape velocity will be ${\left(kg\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$.

Hence, option A is the correct answer.