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JEE MainPhysicsGravitation

Satellites are orbiting very close to the surfaces of two planets A and B . The ratio of the time periods of these satellites is T_A : T_B = 3 : 1 . If the ratio of the surface accelerations due to gravity on the planets is g_A : g_B = 2 : 1 , then the ratio of the escape velocities from their surfaces ( v_ eA : v_ eB ) will be:

Options

  1. A2 : 3
  2. B6 : 1
  3. C6 : 1
  4. D2 : 3

Correct answer

C. 6 : 1

Step-by-step solution

The time period of a satellite orbiting very close to the surface of a planet is given by: T = 2 R g Squaring both sides, we can express the radius R as: R = gT^2 4 ^2 R gT^2 The escape velocity from the surface of a planet is: v_e = 2gR Substituting the proportionality for R into the escape velocity equation: v_e g(gT^2) = gT Taking the ratio for planets A and B : v_ eA v_ eB = ( g_A g_B ) ( T_A T_B ) Given g_A g_B = 2 and T_A T_B = 3 : v_ eA v_ eB = 2 3 = 6 Thus, the ratio of their escape velocities is 6 : 1 . An

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