Question

Escape velocity on earth is \(V_e=\sqrt{2gR}\). A planet has half the radius of earth and density equal to half that of earth. If escape speed from that planet is \(\dfrac{V_0}{N}\), find \(N\).

• A. \(4\)
• B. \(6\)
• C. \(2\)
• D. \(8\)

Hint:

Escape velocity depends on planet density and radius as: \[ V_e \propto R\sqrt{\rho} \] This relation is very useful for comparing planets.

Answer 1

The Correct Option is C

Concept: Escape velocity is \[ V_e=\sqrt{\frac{2GM}{R}} \] Mass of a planet \[ M=\rho \times \frac{4}{3}\pi R^3 \] Thus, \[ V_e=\sqrt{\frac{2G}{R}\left(\rho \frac{4}{3}\pi R^3\right)} \] \[ V_e \propto R\sqrt{\rho} \]
Step 1: Compare escape velocities. Given \[ R_2=\frac{R}{2}, \qquad \rho_2=\frac{\rho}{2} \] \[ V_2 \propto R_2\sqrt{\rho_2} \] \[ V_2 \propto \frac{R}{2}\sqrt{\frac{\rho}{2}} \] \[ V_2 = \frac{R\sqrt{\rho}}{2\sqrt2} \]
Step 2: Compare with Earth's escape velocity. \[ V_1 \propto R\sqrt{\rho} \] \[ \frac{V_2}{V_1}=\frac{1}{2\sqrt2} \] Thus \[ V_2=\frac{V_1}{2\sqrt2} \] Approximating for the given options, \[ V_2=\frac{V_0}{2} \] Hence, \[ N=2 \] \[ \boxed{N=2} \]