Gravitation Ques 4
- A satellite is revolving in a circular orbit at a height $h$ from the earth surface such that $h \ll R$, where $R$ is the radius of the earth. Assuming that the effect of earth’s atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is
(2019 Main, 11 Jan II)
(a) $\sqrt{\frac{g R}{2}}$
(b) $\sqrt{g R}$
(c) $\sqrt{2 g R}$
(d) $\sqrt{g R}(\sqrt{2}-1)$
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Answer:
Correct Answer: 4.( d )
Solution:
- Orbital velocity of the satellite is given as,
$ v_O=\sqrt{\frac{G M}{R+h}} $
Since, $R»h$
$ \therefore \quad v_O=\sqrt{\frac{G M}{R}}=\sqrt{g R} \quad\left[\because g=\frac{G M}{R^2}\right] $
Escape velocity of the satellite,
$ v_c=\sqrt{\frac{2 G M}{R+h}}=\sqrt{\frac{2 G M}{R}}=\sqrt{2 g R} $
Since, we know that in order to escape the earth’s gravitational field a satellite must get escape velocity.
$\therefore$ Change in velocity,
$ \begin{aligned} \Delta v=v_e-v_O & \\ =\sqrt{g R}(\sqrt{2}-1) \end{aligned} $
BETA
