Question

The escape velocity on a planet is equal to the orbital velocity on another planet . If the radii of two planets are equal, determine the ratio of acceleration due to gravity on the two planets

• A. $5:2$
• B. $3:4$
• C. $2:3$
• D. $1:2$

Answer 1

The correct option is D $1:2$

Escape velocity of planet A = $v_E$ = Orbital velocity of planet B=$v_0$

Now, Radius of planet A = Radius of planet B=R

$v_E=\sqrt{\frac{2G{M}_A}{R}}=\sqrt{2g_A/R}{v}_0=\sqrt{\frac{2G{M}_B}{R}}=\sqrt{2g_B/R}{g}_A=\frac{V_E^{2}R}{2}{g}_B=V_0^{2}R$

Now $v_E=v_0=v$(say)

$\frac{g_A}{g_B}=\frac{v_E^2}{2}.\frac{1}{v_0^2}=\frac{v^2}{2}.\frac{1}{v^2}=\frac{1}{2}$

ratio of acc due to gravity on the two planets is 1:2