Question

Let (Vo) and (Ve) represent the orbital velocity and escape velocity of a satellite corresponding to a circular orbit of radius R. Derive the relation between them
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Answer 1

Using the Conservation of Energy, we can derive the Escape Velocity as:

$\sf U_{i} + K_{i} = U_{f} + K_{f}$

$\sf - \frac{GMm}{R} + \frac{1}{2} mv _{esc}^{2} = -\frac{GMm}{h} + 0$

$\because h \to \infty$

$\boxed{ \sf v_{esc} = \sqrt{ \frac{2G M_{earth}}{R_{earth}} } }$

Now, For Orbital Velocity

$\sf F_{gravitational} = F_{centripetal}$

$\sf \frac{mv_{o}^{2} }{2} = \frac{GMm}{R}$

$\boxed{ \sf v _{o} = \sqrt{ \frac{GM_{earth} }{R} } }$

Dividing Both The Equations:

$\sf \frac{v_{esc} }{v_{o}} = \sqrt{ \frac{2R}{R_{earth} } }$

$\implies \boxed{ \sf v_{esc} = v_{o} \sqrt{ \frac{2R}{R_{earth}} } }$

If R=Radius of Earth

$\boxed{ \sf v_{e} = \sqrt{2} v_{o}}$

Answer 2

Answer:

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Explanation:

have a great day ahead ✌️✌️✌️ Rohan