The gravitational potential energy required to raise a satellite of mass ' m ' to height ' h ' above the earth's surface is ' E 1 '. Let the energy required to put

Question

The gravitational potential energy required to raise a satellite of mass ' m ' to height ' h ' above the earth's surface is ' E 1 ​ '. Let the energy required to put this satellite into the orbit at the same height be ' E 2 ​ '. If M and R are the mass and radius of the earth respectively then E 1 ​ : E 2 ​ is

TestHub · Accepted Answer

The gravitational potential energy required to raise the satellite to a height $h$ is the change in its potential energy:

$E_{1} = Δ U = U_{f} - U_{i}$

$E_{1} = - \frac{GM m}{R + h} - (- \frac{GM m}{R})$

$E_{1} = \frac{GM m}{R} - \frac{GM m}{R + h} = \frac{GM mh}{R ( R + h )}$

The energy required to put the satellite into orbit at this height is the kinetic energy needed for orbital motion:

$E_{2} = \frac{1}{2} m v_{o}^{2}$

Since the orbital velocity is $v_{o} = \frac{GM}{R + h}$ , we get:

$E_{2} = \frac{1}{2} m (\frac{GM}{R + h}) = \frac{GM m}{2 ( R + h )}$

Taking the ratio of $E_{1}$ to $E_{2}$ :

$\frac{E _{1}}{E _{2}} = \frac{\frac{GM mh}{R ( R + h )}}{\frac{GM m}{2 ( R + h )}} = \frac{2 h}{R}$

Answer: $2 h : R$

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