Two particles of masses $m$ and $M$ are initially at rest at an infinite distance apart. They move towards each other and gain speeds due to gravitational attraction. Find their speeds when the separation between the masses becomes equal to $d$ .

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Solution

Let $v_{1}$ and $v_{2}$ be the speeds of two masses m and M, respectively, when they are at a separation d.
As they approach each other, the kinetic energy increases and GPE decreases. Hence, for the system,
Loss in GPE $=$ Gain in KE
$\Rightarrow (G P E)_{i} - (G P E)_{f} - K E_{f} - K E_{i}$
$\Rightarrow 0 - (- \frac{G M m}{d}) = (\frac{1}{2} m v_{1}^{2} + \frac{1}{2} M v_{2}^{2}) - 0$
$\frac{G M m}{d} = \frac{1}{2} m v_{1}^{2} + \frac{1}{2} M v_{2}^{2}$
As there is no external force on this system, its total momentum remains conserved.
$P_{i} = P_{f}$
$0 = m v_{1} - M v_{2}$
Combining the two equations, we have
$v_{1} = \sqrt{\frac{2 G M^{2}}{d (m + M)}}$
and $v_{2} = \sqrt{\frac{2 G M^{2}}{d (m + M)}}$ .

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Two particles of masses m and M are initially at rest at an infinite distance apart. They move towards each other and gain speeds due to gravitational attraction. Find their speeds when the separation between the masses becomes equal to d.

Two particles of masses $m$ and $M$ are initially at rest at an infinite distance apart. They move towards each other and gain speeds due to gravitational attraction. Find their speeds when the separation between the masses becomes equal to $d$ .