Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The escape velocities of planet A and B are same. But A and B are of unequal mass.
Reason R : The product of their mass and radius must be same. M₁R₁ = M₂R₂
In the light of the above statements, choose the most appropriate answer from the options given below :

Solution

Physics sir · 4 steps

Escape velocity depends on M divided by R, so equal escape speeds need equal M/R, not equal MR.

1. Escape velocity formula. Escape speed comes from setting the kinetic energy 1/2mv² equal to the gravitational binding energy GMm/R at the surface.

   vₑ = √(2GM/R)

2. Check Assertion A. If vₑ₁ = vₑ₂, then M₁/R₁ = M₂/R₂. Two planets with different masses can satisfy this if their radii are in the same ratio, for example M₂ = 2M₁ and R₂ = 2R₁. So A is correct.
3. Check Reason R. R says M₁R₁ = M₂R₂. With M₂ = 2M₁ and R₂ = 2R₁, the product M₂R₂ = 4M₁R₁, which is not equal to M₁R₁. The condition that actually holds is equal M/R, so R is wrong.
4. So. A is correct but R is not correct, which is option (c).

The idea to keep. For equal escape speeds the ratio M/R must match. If the planets have the same density, vₑ grows with R, because M/R grows as R².

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