Question

The additional kinetic energy to be provided to a satellite of mass $m$ revolving around a planet of mass $M$ to transfer it from a circular orbit of radius $R_1$ to another circular orbit of radius $R_2(R_2>R_1)$ is

• A. $GMm(\frac{1}{R_1}+\frac{1}{R_2})$
• B. $GMm(\frac{1}{R_1}-\frac{1}{R_2})$
• C. $2GMm(\frac{1}{R_1}-\frac{1}{R_2})$
• D. $\frac{1}{2}GMm(\frac{1}{R_1}-\frac{1}{R_2})$

Answer 1

The correct option is D $\frac{1}{2}GMm(\frac{1}{R_1}-\frac{1}{R_2})$

Total mechanical energy of a satellite in circular orbit is given by $$E=-\frac{GMm}{2r}$$Where $r$ is the radius of the orbit.

So, the total energy of the satellite in the orbit of radius $R_1$ and $R_2$ are respectively:

$E_1=\frac{-GMm}{2R_1}$ and $E_2=\frac{-GMm}{2R_2}$

The additional kinetic energy required will be the difference between the total mechanical energy in the two orbits.

Thus, additional kinetic energy:

$K.E=E_2-E_1$

$\Rightarrow K.E=\frac{-GMm}{2R_2}-\frac{-GMm}{2R_1}$

$\Rightarrow K.E=\frac{1}{2}GMm(\frac{1}{R_1}-\frac{1}{R_2})$

Hence, option (d) is the correct answer.

Key Concept: The kinetic energy required to shift a satellite from one orbit to the other is equal to the difference between the total mechanical energy of the two configurations.