Question

The escape velocity for earth is v. A planet having 9 times mass that of earth and radius, 16 times that of earth, has the escape velocity of:

(1) \(\frac{v}{3}\)

(2) \(\frac{2 v}{3}\)

(3) \(\frac{3 v}{4}\)

(4) \(\frac{9 v}{4}\)

Answer 1

Correct option is : (3) \(\frac{3 v}{4}\) 

Escape velocity of object from planet is given by

\(\left(v_{e}\right)_{p}=\sqrt{\frac{2 G M_{p}}{R_{p}}}\)

\(\therefore \quad\left(v_{e}\right)_{p} \propto \sqrt{\frac{M_{p}}{R_{p}}}\)

Now, \(M_{p}=9 M_{e}\) and \(R_{p}=16 R_{e}\) (given)

\(\frac{\left(v_{e}\right)_{p}}{\left(v_{e}\right)_{e}}=\sqrt{\frac{M_{p}}{R_{p}} \times \frac{R_{e}}{M_{e}}}=\sqrt{\frac{9 M_{e}}{16 R_{e}} \times \frac{R_{e}}{M_{e}}}=\sqrt{\frac{9}{16}}=\frac{3}{4}\)

\(\Rightarrow \quad\left(v_{e}\right)_{p}=\frac{3}{4} v\)