The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is ____ m/s.

Solution

Physics sir · 4 steps

Write escape velocity in terms of density, and you see it scales as R√ρ. Then just scale.

1. Replace mass. Escape velocity is vₑ = √(2GM/R). Put in M = 4/3π R³ ρ so that density appears instead of mass.

   vₑ = √(2G/R·4/3π R³ ρ) = R√((8π Gρ)/3) ⇒ vₑ ∝ R√ρ

2. Take the ratio. For planet B, R_B = 0.1R_A and ρ_B = 0.1ρ_A. So v_B/v_A = 0.1 × √0.1 = 1/10√10.
3. Put in numbers. v_A = 10 km/s = 10⁴ m/s, so v_B = 10⁴/10√10 = 1000/√10 = 100√10 m/s, about 316 m/s.
4. So. The escape velocity from planet B is 100√10 m/s, which is option (d).

The idea to keep. When a question gives density instead of mass, write M = 4/3π R³ρ first. Escape velocity then goes as R√ρ, and surface gravity g goes as Rρ.

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