Two bodies of masses m 1 and m 2 are initially at rest placed infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity when they are r distance apart is

Question

Two bodies of masses m 1 and m 2 are initially at rest placed infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity when they are r distance apart is, 2 G ( m 1 + m 2 ) r, 2 G m 1 m 2 ( m 1 + m 2 ) r, G ( m 1 + m 2 ) r, G m 1 m 2 m 1 + m 2 r

MARKS App · Accepted Answer

By conservation of momentum m 1 v 1 - m 2 v 2 = 0 by conservation of energy 1 2 m 1 v 1 2 + 1 2 m 2 v 2 2 - G m 1 m 2 r = 0 1 2 m 1 v 1 2 + 1 2 m 2 m 1 v 1 m 2 2 = G m 1 m 2 r v 1 2 2 m 1 + m 1 2 m 2 = G m 1 m 2 r v 1 2 = 2 G m 2 2 m 1 + m 2 r v 1 = m 2 2 G m 1 + m 2 r ∴ v 2 = m 1 2 G m 1 + m 2 r Also, v r e l = v 1 + v 2 v r e l = m 2 + m 1 2 G m 1 + m 2 r v r e l = 2 G m 1 + m 2 r

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