Problem 2 - A Complete Resource Book in Physics for JEE Main

Question

The ratio of the radii of the planets \(P_{1}\) and \(P_{2}\) is \(k_{1}\). The ratio of the acceleration due to the gravity on them is \(k_{2}\). The ratio of the escape velocities from them will be (A) \(k_{1} k_{2}\) (B) \(\sqrt{k_{1} k_{2}}\) (C) \(\sqrt{\left(k_{1} / k_{2}\right)}\) (D) \(\sqrt{\left(k_{2} / k_{1}\right)}\)

Step-by-Step Solution

Answer

The ratio of the escape velocities from planets \(P_1\) and \(P_2\) is \(\sqrt{k_1 k_2}\).

Step 1: Write the formula for escape velocity

The escape velocity (v) of a celestial body can be calculated using the formula: \( v = \sqrt{2 \times g \times R}\) Where g is the acceleration due to gravity on the body, and R is the radius of the body.

Step 2: Write the given values and ratios

We are given that the ratio of radii of planets P1 and P2 is \(k_1\), and the ratio of acceleration due to gravity on them is \(k_2\). Let the radius of planet P1 be \(R_1\), and P2 be \(R_2\), so \(R_1 = k_1 R_2\) Also, let the acceleration due to gravity on planet P1 be \(g_1\), and P2 be \(g_2\), so \(g_1 = k_2 g_2\)

Step 3: Write the escape velocities for both planets

Using the formula for escape velocity, the escape velocities of planets P1 and P2 can be calculated as: Escape velocity of planet P1: \( v_1 = \sqrt{2 \times g_1 \times R_1} \) Escape velocity of planet P2: \( v_2 = \sqrt{2 \times g_2 \times R_2} \)

Step 4: Find the ratio of escape velocities

To find the ratio of escape velocities, divide \(v_1\) by \(v_2\): \(\frac{v_1}{v_2} = \frac{\sqrt{2 \times g_1 \times R_1}}{\sqrt{2 \times g_2 \times R_2}}\) Substitute values of \(g_1\) and \(R_1\) \(\frac{v_1}{v_2} = \frac{\sqrt{2 \times (k_2 g_2) \times (k_1 R_2)}}{\sqrt{2 \times g_2 \times R_2}}\) Simplify the equation: \(\frac{v_1}{v_2} = \frac{\sqrt{k_1 k_2 \times g_2 \times R_2}}{\sqrt{g_2 \times R_2}}\) \(\frac{v_1}{v_2} = \frac{\sqrt{k_1 k_2}}{1}\) \(\frac{v_1}{v_2} = \sqrt{k_1 k_2}\) So, the ratio of the escape velocities from planets P1 and P2 is \(\sqrt{k_1 k_2}\). Hence, the correct option is (B).