Problem 84 - Master Resource Book in JEE Main Physics

Question

The ratio of the radii of the planets \(P_{1}\) and \(P_{2}\) is \(a\). The ratio of their acceleration due to gravity is \(b .\) The ratio of the escape velocities from them will be (a) \(a b\) (b) \(\sqrt{a b}\) (c) \(\sqrt{a / b}\) (d) \(\sqrt{b / a}\)

Step-by-Step Solution

Answer

The ratio of the escape velocities is \(\sqrt{ab}\), option (b).

Step 1: Understanding Given Ratios

We know the ratio of the radii of planets \(P_1\) and \(P_2\) is \(a\), meaning \(\frac{R_1}{R_2} = a\). The ratio of the acceleration due to gravity is \(b\), meaning \(\frac{g_1}{g_2} = b\). We need to derive the formula for the ratio of escape velocities.

Step 2: Escape Velocity Formula

The escape velocity \(v_e\) from a planet is given by the formula:\[v_e = \sqrt{2gR}\].This means the escape velocity for \(P_1\) is \(v_{e1} = \sqrt{2g_1R_1}\) and for \(P_2\) is \(v_{e2} = \sqrt{2g_2R_2}\).

Step 3: Setting Up the Ratio for Escape Velocities

The ratio of escape velocities \(\frac{v_{e1}}{v_{e2}}\) is given by:\[\frac{v_{e1}}{v_{e2}} = \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \sqrt{\frac{g_1R_1}{g_2R_2}}\].

Step 4: Substituting Known Ratios

Substitute the known ratios \(\frac{g_1}{g_2} = b\) and \(\frac{R_1}{R_2} = a\) into the escape velocity ratio:\[\sqrt{\frac{g_1R_1}{g_2R_2}} = \sqrt{\frac{g_1}{g_2} \cdot \frac{R_1}{R_2}} = \sqrt{b \cdot a}\].

Step 5: Conclusion

Thus, the ratio of the escape velocities from the planets \(P_1\) and \(P_2\) is \(\sqrt{ab}\). Hence, the correct answer is option (b).

Key Concepts

Acceleration Due to Gravity

Acceleration due to gravity is a fundamental force that pulls objects towards the center of a planet. This force is unique for each celestial body and usually represented by the symbol \(g\). It varies based on several factors:

• The mass of the planet: Larger masses generate stronger gravitational forces.
• The radius of the planet: Gravity is inversely proportional to the square of the radius.

To understand how gravity differs between two planets, if planet \(P_1\) has a ratio \(b\) times the gravity of planet \(P_2\), it tells us how the force pulling objects down on each planet compares. This ratio plays a crucial role when calculating escape velocities, as seen later.

Planetary Radii

Planetary radii determine several dynamical properties of a planet, including its gravitational force and escape velocity. The radius \(R\) is the distance from the planet's center to its surface. When comparing two planets such as \(P_1\) and \(P_2\), the ratio of their radii \(a = \frac{R_1}{R_2}\) helps understand their relative sizes.

• A larger radius would suggest more surface area from which gravity acts.
• A smaller radius generally implies a stronger surface gravity, assuming similar masses.

Thus, understanding planetary radii can aid in comparing the escape velocities of different celestial bodies, as it directly influences the gravitational energy potential needed to escape a planet's gravitational pull.

Velocity Ratio

The velocity ratio between two planets is pivotal in determining how fast an object must travel to escape from the gravitational pull of each planet.
Escape velocity \(v_e\) is derived from the principles of gravitational force and is defined by the formula \(v_e = \sqrt{2gR}\). This incorporates both the acceleration due to gravity \(g\) and the planetary radius \(R\).

• The ratio of escape velocities \(\frac{v_{e1}}{v_{e2}}\) for two planets shows how diverging parameters like gravity and size affect their gravitational thresholds.
• In the exercise, substituting \(\frac{g_1}{g_2} = b\) and \(\frac{R_1}{R_2} = a\) into the equation gives us the velocity ratio as \(\sqrt{ab}\).

Therefore, the ratio \(\sqrt{ab}\) reveals the relationship of gravitational forces and physical dimensions between planets, directly impacting what speed is required for space travel from each one.