Question

The ratio of radii of two planets is $p$ and the ratio of acceleration due to gravity at their surfaces is $q$. The ratio of escape velocities from their surfaces will be

• A. ${\left[\frac{q}{p}\right]}^{1/2}$
• B. ${\left[\frac{p}{q}\right]}^{1/2}$
• C. $pq$
• D. $\sqrt{pq}$

Answer 1

The correct option is D $\sqrt{pq}$

Escape velocity, $v_e=\sqrt{\frac{2GM}{R}}$

Using $GM=g{R}^2$ where $g$ is acceleration due to gravity at surface.

$⟹v_e=\sqrt{2gR}$

Now $\frac{v_e^{\prime \prime }}{v_e^{\prime }}=\frac{\sqrt{2g^{\prime \prime }{R}^{\prime \prime }}}{\sqrt{2g^{\prime }{R}^{\prime }}}=\sqrt{\frac{g^{\prime \prime }}{g^{\prime }}\frac{R^{\prime \prime }}{R^{\prime }}}$

$⟹\frac{v_e^{\prime \prime }}{v_e^{\prime }}=\sqrt{qp}$