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JEE MainPhysicsGravitation

Two planets A and B have surface accelerations due to gravity in the ratio 2 : 1 and average mass densities in the ratio 1 : 4 . The ratio of the escape velocities from the surfaces of these planets ( v_ eA : v_ eB ) is:

Options

  1. A1 : 2
  2. B4 : 1
  3. C1 : 2
  4. D2 2 : 1

Correct answer

B. 4 : 1

Step-by-step solution

The acceleration due to gravity on the surface of a planet is given by: g = GM R^2 = G ( 4 3 R^3 ) R^2 = 4 3 G R From this, the radius R can be expressed as: R g The escape velocity from the surface is: v_e = 2gR Substituting the proportionality for R : v_e g ( g ) = g Taking the ratio for planets A and B : v_ eA v_ eB = g_A g_B _B _A Given g_A g_B = 2 and _A _B = 1 4 : v_ eA v_ eB = 2 4 1 = 2 2 = 4 Thus, the ratio is 4 : 1 . Answer: 4 : 1

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