First slide

Gravitaional potential energy

Question

The escape velocity for a planet is $\large \upsilon _e$ . A particle is projected horizontally from the top of tower of height h with a speed $\large \upsilon$ . For this particle to move as a satellite around the planet

Moderate

A

$\large \frac{{{\upsilon _e}}}{2} < \upsilon < {\upsilon _e}$
B

$\large \frac{{{\upsilon _e}}}{{\sqrt 2 }} < \upsilon < {\upsilon _e}$
C

$\large {\upsilon _e} < \upsilon < \sqrt 2 {\upsilon _e}$
D

$\large \frac{{{\upsilon _e}}}{{\sqrt 2 }} < \upsilon < \frac{{{\upsilon _e}}}{2}$

Solution

$\large \upsilon_e$ = Escape velocity of planet = $\large \sqrt{\frac{2GM}{R+h}}=\sqrt{\frac{2GM}{r}}$ .If $\large \upsilon \geq \upsilon_e$ .The particle will escape to infinity following parabolic or hyperbolic path . If $\large \upsilon = \frac{\upsilon _e}{\sqrt2}=\sqrt{\frac{GM}{r}}$ , the particle will follow a circular path around the planet.If $\large {\frac{\upsilon_e}{\sqrt2}}<\upsilon <\upsilon _e$ , the particle will follow an elliptical path around the planet.If $\large \upsilon<\frac {\upsilon_e}{\sqrt2}$ , the particle will follow an elliptical path and strike the surface the earth.

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