The escape velocity of a body on a planet 'A' is 12 kms⁻¹. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is :

Solution

Physics sir · 3 steps

Write the mass in terms of density, and escape velocity depends only on R times the square root of density.

1. Bring in density. Escape velocity is vₑ = √(2GM/R). For a uniform planet, M = 4/3π R³ ρ. Put this in and one R cancels.

   vₑ = √((8π G ρ)/3) R ⇒ vₑ ∝ R√ρ

2. Take the ratio. For planet B, R_B = 1/2R_A and ρ_B = 4ρ_A. So v_B/v_A = R_B/R_A√(ρ_B/ρ_A) = 1/2 × √4 = 1.
3. So. The halved radius and the doubled √ρ cancel exactly. That gives v_B = v_A = 12 km s⁻¹, option (a).

The idea to keep. When a question gives density instead of mass, rewrite escape velocity as vₑ ∝ R√ρ. Then change both factors together before you multiply.

Answer from the NTA final answer key. Solution by PrepWiser.