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  • The masses and radii of the Earth and the Moon are <img alt="\mathrm{M}_1, \mathrm{R}_1 \text { and } \mathrm{M}_2, \mathrm{R}_2" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BM%

The masses and radii of the Earth and the Moon are \mathrm{M}_1, \mathrm{R}_1 \text { and } \mathrm{M}_2, \mathrm{R}_2 respectively. Their centers are a distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway between the two centers so as to escape to infinity is ………………

Option: 1

\sqrt{\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{2 \mathrm{~d}}}


Option: 2

\sqrt{\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{ \mathrm{~d}}}


Option: 3

\sqrt{\frac{\mathrm{2G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{2 \mathrm{~d}}}


Option: 4

2\sqrt{\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{2 \mathrm{~d}}}


Answers (1)

best_answer

Total mechanical energy of mass m at appoint midway between two centers is :

\mathrm{\begin{aligned} \mathrm{E} & =-\frac{G M_1 m}{d / 2}-\frac{G M_2 m}{d / 2} \\ & =-\frac{2 G M}{d}\left(\mathrm{M}_1+\mathrm{M}_2\right) \end{aligned}}

\mathrm{\text { Binding energy }=\frac{2 G m}{d}\left(\mathrm{M}_1+\mathrm{M}_2\right)}

Kinetic energy required to escape the mass to infinity is,

\mathrm{\begin{aligned} \frac{1}{2} m v_e^2 & =\frac{2 G m}{d}\left(\mathrm{M}_1+\mathrm{M}_2\right) \\ \therefore \quad \quad \quad \mathrm{v}_{\mathrm{e}} & =2 \sqrt{\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{\mathrm{d}}} \end{aligned}}

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jitender.kumar

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