The mass and radius of the earth and moon are | vedclass

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Question

(A) The gravitational potential energy $U$ of a particle of mass $m_0$ at the midpoint between the earth and the moon is given by the sum of potential

Vedclass · Accepted Answer

$2 \sqrt{\frac{G}{d}(M+m)}$

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(A) The gravitational potential energy $U$ of a particle of mass $m_0$ at the midpoint between the earth and the moon is given by the sum of potentials due to both bodies:$U = -\frac{G M m_0}{d/2} - \frac{G m m_0}{d/2} = -\frac{2 G m_0}{d} (M + m)$To reach infinity,the total energy of the particle must be at least zero. Let $V$ be the minimum projection velocity.Applying the principle of conservation of energy:$K_i + U_i = K_f + U_f$$\frac{1}{2} m_0 V^2 - \frac{2 G m_0}{d} (M + m) = 0 + 0$$\frac{1}{2} m_0 V^2 = \frac{2 G m_0}{d} (M + m)$$V^2 = \frac{4 G}{d} (M + m)$$V = 2 \sqrt{\frac{G(M+m)}{d}}$

The mass and radius of the earth and moon are $M, R$ and $m, r$ respectively. The distance between their centers is $d$. The minimum velocity with which a particle of mass $m_0$ should be projected from the midpoint between them so that it could reach infinity is

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