Two planets A and B of masses mA and mB 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 $$nGmA+mBd$$. Find the value of n.

Question

Two planets A and B of masses  mA and mB 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 $$nGmA+mBd$$. Find the value of n.

Infinity Learn · Accepted Answer

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

$T E \geq 0 \frac{1}{2} m v^{2} + m v_{0} \geq 0 \frac{1}{2} v^{2} + v_{0} \geq 0 \frac{1}{2} v^{2} - \frac{G m_{A}}{d / 2} - \frac{G m_{B}}{d / 2} \geq 0 v \geq \sqrt{\frac{4 G (m_{A} + m_{B})}{d}} ∴ n = 4$

Motion of planets and satellites

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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.

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Motion of planets and satellites

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Two planets A and B of masses mA and mB 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 nGmA+mBd. Find the value of n.

Due to planets A and B, gravitational potential at mid point V0=-GmAd/2-GmBd/2For body projected from O to escape to infinity TE≥0 12mv2+mv0≥0 12v2+v0≥0 12v2-GmAd/2-GmBd/2≥0 v≥4GmA+mBd ∴ n=4

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free study material

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.