Question

Planet $$A$$ has mass $$M$$ and radius $$R$$. Planet $$B$$ has half the mass and half the radius of planet $$A$$. If the escape velocities from the planets $$A$$ and $$B$$ are $$\upsilon _A$$ and $$\upsilon _B$$ respectively, then $$\dfrac {\upsilon _A}{\upsilon _B} =\dfrac {n}{4}$$.

The value of $$n$$ is

Answer 1

Correct option is C. $$4$$
$$V_{esc}=$$ escape velocity $$=\sqrt{\dfrac{2GM}{R}}$$

$$V_A=\sqrt{\dfrac{2GM}{R}}$$

$$V_B=\sqrt{\dfrac{2G\times \left(\dfrac{M}{2}\right)}{(R/2)}}=\sqrt{\dfrac{2GM}{R}}$$

$$V_A=V_B$$

$$\dfrac{V_A}{V_B}=1=\dfrac{n}{4}$$

$$\therefore \boxed{n=4}$$