The masses and radii of the Earth and Moon ar | vedclass

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(A) The gravitational potential $V$ at the midpoint between the centers of the Earth and the Moon is given by the sum of potentials due to both masses

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$2\sqrt{\frac{G}{d}(M_1 + M_2)}$

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(A) The gravitational potential $V$ at the midpoint between the centers of the Earth and the Moon is given by the sum of potentials due to both masses:$V = -\frac{GM_1}{d/2} - \frac{GM_2}{d/2} = -\frac{2G}{d}(M_1 + M_2)$The potential energy $PE$ of a particle of mass $m$ at this point is:$PE = m 	imes V = -\frac{2G m}{d}(M_1 + M_2)$To escape to infinity,the total energy of the particle must be at least zero. Therefore,the kinetic energy $KE$ provided must equal the magnitude of the potential energy:$KE = |PE|$$\frac{1}{2}mv^2 = \frac{2G m}{d}(M_1 + M_2)$Solving for the velocity $v$:$v^2 = \frac{4G}{d}(M_1 + M_2)$$v = 2\sqrt{\frac{G}{d}(M_1 + M_2)}$

The masses and radii of the Earth and Moon are $M_1, R_1$ and $M_2, R_2$ respectively. Their centers are at a distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centers so that it escapes to infinity is:

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