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JEE MainPhysicsGravitation

Two spherical planets, A and B, have uniform mass densities. Planet B has a uniform density that is 4 times the density of planet A, and its radius is half the radius of planet A. If the escape velocity from the surface of planet A is v , then the escape velocity from the surface of planet B is

Options

  1. A2 2 v
  2. B2v
  3. Cv
  4. Dv 2

Correct answer

C. v

Step-by-step solution

The escape velocity from the surface of a planet of mass M and radius R is given by v = 2GM R Expressing the mass in terms of uniform density , we have M = 4 3 R^3 . Substituting this into the escape velocity formula gives: v = 2G R ( 4 3 R^3 ) = R 8 G 3 This shows that the escape velocity is directly proportional to the radius and the square root of the density: v R For planets A and B, we can write the ratio of their escape velocities as: v_B v_A = ( R_B R_A ) _B _A Given that R_B = 1 2 R_A and _B = 4 _A , we sub

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