Question

Two satellites of identical masses orbit the earth at different heights. The ratio of their distances from the centre of earth is d: 1 and the ratio of the acceleration due to gravity at those heights is g: 1. Then find the ratio of their orbital velocities.

Answer 1

Let mass of the earth is M and the radius of the earth is r.

ratio of their heights from the centre of the earth is d : 1. means, if 2nd body is at r distance from the centre of earth then, 1st body is at rd distance from the centre of the earth.

similarly, ratio of the acceleration due to gravity at those heights is g : 1. means, if the acceleration due to gravity for 1st body is g then, acceleration due to gravity for 2nd body is 1.

now, use formula, orbital velocity = $\bf{\sqrt{gr}}$

here, g is acceleration due to gravity experienced by object at r distance of object from the centre of the earth.

now, ratio of their orbital velocities = $\bf{\frac{\sqrt{g.rd}}{\sqrt{1.r}}}=\bf{\sqrt{dg}}$