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93. Problem 14.53P (HRW) The gravitational force between two particles with mass m and M, initially at
rest at great separation, pulls them together. We have to show that at any
instant the speed of either particle relative to the other is
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where d is their separation at that instant.|
Solution: Click For PDF Version According to the problem when the two particles of mass m and M are infinitely separated their kinetic energy and the mutual gravitational potential energy is zero. That is their energy
Let the speeds of the particles m and M, when their mutual separation distance is d, be v and V, respectively. At that instant the speed of either particle relative to the other will be v +V. As the initial momentum of the two particle system is zero, so by conservation of momentum at any instant
From conservation of energy at any instant
Solving for v and V, we get
Therefore, at any instant the speed of the either particle relative to the other is
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