Relative Velocity of Approaching Particles – Rankers Physics
Topic: Gravitation
Subtopic: Gravitational Potential Energy

Relative Velocity of Approaching Particles

The gravitational force between two particles with masses \(m\) and \(M\), initially at rest at great separation, pulls them together. When their separation becomes \(d\), then speed of either particle relative to the other will be :
\(\sqrt{G(M+m)/2d}\)
\(\sqrt{G(M+m)/d}\)
\(\sqrt{4G(M+m)/d}\)
\(\sqrt{2G(M+m)/d}\)

Solution:

By conservation of mechanical energy, the relative speed is found using the reduced mass \(\mu = \frac{mM}{m+M}\). Thus, \(\frac{1}{2} mu v_{\text{rel}}^2 = \frac{GMm}{d}\), which simplifies to \(v_{\text{rel}} = \sqrt{\frac{2G(M+m)}{d}}\).

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