The surface gravitational acceleration of a planet is g/11, where g is Earth's surface gravitational acceleration. The planet's mean density is one-fourth of Earth's. If Earth's escape speed is 11 km s⁻¹, find the planet's escape speed in km s⁻¹.
Physics · Gravitation · Acceleration Due to Gravity & Escape Velocity
JEE Mains Advanced

Question

The surface gravitational acceleration of a planet is g/11, where g is Earth's surface gravitational acceleration. The planet's mean density is one-fourth of Earth's. If Earth's escape speed is 11 km s⁻¹, find the planet's escape speed in km s⁻¹.

Physics · Gravitation · Acceleration Due to Gravity & Escape Velocity

G1 Physics · Accepted Answer

For a spherical body, g ∝ ρR. Thus (gₚ/gₑ) = (ρₚ/ρₑ)(Rₚ/Rₑ), so 1/11 = (1/4)(Rₚ/Rₑ), giving Rₚ/Rₑ = 4/11. Since vₑₛc² = 2gR, (vₚ/vₑ)² = (1/11)(4/11) = 4/121. Therefore vₚ = (2/11)(11 km s⁻¹) = 2 km s⁻¹.