By how many times the energy of a satellite has to be increased to shift it from an orbit of radius 'r' to 2 5 r ?

Question

By how many times the energy of a satellite has to be increased to shift it from an orbit of radius 'r' to 2 5 ​ r ?

TestHub · Accepted Answer

The total energy of a satellite in an orbit of radius $r$ is given by:

$E_{1} = - \frac{GM m}{2 r}$

The total energy of the satellite in an orbit of radius $\frac{5}{2} r$ is:

$E_{2} = - \frac{GM m}{2 ( \frac{5}{2} r )} = - \frac{GM m}{5 r}$

The increase in energy required to shift the satellite is:

$Δ E = E_{2} - E_{1}$

$Δ E = - \frac{GM m}{5 r} - (- \frac{GM m}{2 r}) = \frac{GM m}{r} (\frac{1}{2} - \frac{1}{5}) = \frac{3 GM m}{10 r}$

The magnitude of the initial total energy is $∣ E_{1} ∣ = \frac{GM m}{2 r}$ .

The ratio of the required increase in energy to the magnitude of the initial energy is:

$\frac{Δ E}{∣ E _{1} ∣} = \frac{\frac{3 GM m}{10 r}}{\frac{GM m}{2 r}} = \frac{6}{10} = 0.6$

Thus, the energy has to be increased by $0.6$ times.

Answer: $0.6$ times

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