The masses and radii of the earth and the moo | vedclass

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Question

$PE$ of particle of mass $m$ placed at a point $mid-way$ between the earth and the moon,

$\mathrm{U}=-\left(\frac{\mathrm{GM}_{1} \mathrm{m}}{\left

Vedclass · Accepted Answer

$v = \sqrt {\frac{{4G({M_1} + {M_2})}}{d}} $

Vedclass · Answer

$PE$ of particle of mass $m$ placed at a point $mid-way$ between the earth and the moon,

$\mathrm{U}=-\left(\frac{\mathrm{GM}_{1} \mathrm{m}}{\left(\frac{\mathrm{d}}{2}\right)}+\frac{\mathrm{GM}_{2} \mathrm{m}}{\left(\frac{\mathrm{d}}{2}\right)}\right)=-\frac{2 \mathrm{Gm}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{d}}$

The particle will escape if its total energy becomes zero, i.e.

$\mathrm{U}+\mathrm{K}=0$

$\Rightarrow-\frac{2 \mathrm{Gm}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{d}}+\frac{1}{2} \mathrm{mv}^{2}=0$

$\Rightarrow \mathrm{v}=\sqrt{\frac{4 \mathrm{G}\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)}{\mathrm{d}}}$

The masses and radii of the earth and the moon are $M_1, R_1$ and $M_2, R_2$ respectively. Their centres are distance $d$ apart. The minimum speed with which particle of mass $m$ should be projected from a point midway between the two centres so as to escape to infinity is

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