JEE Main 2023 · 13 April · evening · Single correct · +4 / −1
Given below are two statements:
Statement I : For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases.
Statement II : Escape velocity is independent of the radius of the planet.
In the light of above statements, choose the most appropriate answer form the options given below
- (a) Both Statement I and Statement II are correct
- (b) Statement I is correct but statement II is incorrect
- (c) Both Statement I and Statement II are incorrect
- (d) Statement I is incorrect but statement II is correct
Answer and solution
Answer (b) Statement I is correct but statement II is incorrect
Solution
Physics sir · 4 steps
Write vₑ = √(2GM/R) and read it: M and R sit together as the ratio M/R, so R does matter.
- Energy condition. To escape, the kinetic energy must cancel the gravitational potential energy at the surface: 1/2mvₑ² = GMm/R. The mass m of the object cancels.
vₑ = √(2GM/R) = √(2G × M/R)
- Check Statement I. vₑ depends on √(M/R). If M/R increases, vₑ increases. Statement I is correct.
- Check Statement II. R appears in the formula, so for a fixed mass, vₑ ∝ 1/√R. A smaller planet of the same mass has a larger escape velocity. Statement II is incorrect.
- So. Statement I is correct and Statement II is incorrect, which is option (b).
The idea to keep. Escape velocity is independent of the escaping body's mass, not of the planet's size. It depends on the planet through M/R, or √2gR if you know g.
Answer from the NTA final answer key. Solution by PrepWiser.