A particle is projected from point A , that is at a distance 4 R from the centre of the earth, with speed v 1 in a direction making 30 ° with the line joining the centre of the earth and point A , as

Question

A particle is projected from point A , that is at a distance 4 R from the centre of the earth, with speed v 1 in a direction making 30 ° with the line joining the centre of the earth and point A , as shown. Find the speed v 1 if particle grazes the surface of the earth as shown in the figure. Consider gravitational interaction only between these two. (use G M R = 6.4 × 10 7 m 2 s - 2 )

TestHub · Accepted Answer

Conserving angular momentum w.r.t centre of earth, ( V 1 cos ⁡ 60 ° ) 4 R = mV 2 R ⇒ V 2 V 1 = 2 Conserving mechanical energy of the system, - G M m 4 R + 1 2 m V 1 2 = - G M m R + 1 2 m V 2 2 1 2 V 2 2 - 1 2 V 1 2 = 3 4 G M R V 1 2 = 1 2 G M R V 1 = 1 2 64 × 10 6 = 8000 2 m s - 1 = 4 2 km s - 1

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JEE Advanced 2027 (Class 12)