Question

The ratio of the radii of the planets $P_1$ and $P_2$ is $k$. The ratio of gravitational field intensity at their surface is $r$ then the ratio of the escape velocities from them will be -

• A. $kr$
• B. $\sqrt{kr}$
• C. $\sqrt{\frac{k}{r}}$
• D. $\sqrt{\frac{r}{k}}$

Answer 1

Answer: (B)

$\sqrt{kr}$

We are given that:

$\frac{\frac{G{M}_1}{R_1^2}}{\frac{G{M}_2}{R_2^2}}=r$

$\frac{R_1}{R_2}=k$.

And the escape velocity is given by: $v=\sqrt{\frac{2GM}{R}}$ where $M$ is the mass of the planet and $R$ is its radius.

So, $\frac{v_1}{v_2}=\sqrt{\frac{M_1}{M_2}/\frac{R_1}{R_2}}=\sqrt{\frac{r{k}^2}{k}}=\sqrt{rk}$.