First slide

Motion of planets and satellites

Question

Two planets A and B of masses m_A and m_B are considered fixed in space at separation d. The speed with which a body of mass m is to be projected from the midpoint of the line joining A and B so that the body will escape to infinity is $\sqrt{\frac{nG (m_{A} + m_{B})}{d}}$ . Find the value of n.

Moderate

Solution

Due to planets A and B, gravitational potential at mid point $V_{0} = - \frac{G m_{A}}{d / 2} - \frac{G m_{B}}{d / 2}$
For body projected from O to escape to infinity
$T E \geq 0 \frac{1}{2} m v^{2} + m v_{0} \geq 0 \frac{1}{2} v^{2} + v_{0} \geq 0 \frac{1}{2} v^{2} - \frac{G m_{A}}{d / 2} - \frac{G m_{B}}{d / 2} \geq 0 v \geq \sqrt{\frac{4 G (m_{A} + m_{B})}{d}} ∴ n = 4$

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