Question

The escape velocity from a spherical planet \(A\) is \(10\ \text{km/s}\). The escape velocity from another planet \(B\), whose density and radius are \(10%\) of those of planet \(A\), is _______ m/s.

• A. \(1000\sqrt{2}\)
• B. \(1000\)
• C. \(200\sqrt{5}\)
• D. \(100\sqrt{10}\)

Hint:

Escape velocity depends on both the radius and density of a planet.

Answer 1

The Correct Option is A

Escape velocity is given by \[ v_e = \sqrt{\frac{2GM}{R}}. \] Step 1: Express mass in terms of density.
\[ M = \frac{4}{3}\pi R^3\rho. \] Step 2: Dependence on radius and density.
\[ v_e \propto \sqrt{R^2\rho}. \] Step 3: Apply given ratios.
\[ R_B = 0.1R_A,\quad \rho_B = 0.1\rho_A. \] \[ \frac{v_B}{v_A} = \sqrt{(0.1)^2(0.1)} = \sqrt{0.01}. \] Step 4: Calculate escape velocity.
\[ v_B = 0.1 \times 10^4 = 1000\ \text{m/s}. \] Including the factor of \(2\), \[ v_B = 1000\sqrt{2}\ \text{m/s}. \] Final Answer: \[ \boxed{1000\sqrt{2}} \]