Question

A satellite of mass $1000\text{kg}$ is rotating around the earth in a circular orbit of radius $3R$. What extra energy should be given to this satellite if it is to be lifted into an orbit of radius $4R$ ?

• A. $2.614\times {10}^9\text{J}$
• B. $6\times {10}^8\text{J}$
• C. $3\times {10}^8\text{J}$
• D. $5.3\times {10}^9\text{J}$

Answer 1

The correct option is A $2.614\times {10}^9\text{J}$

Given:
Satellite of mass $1000\text{kg}$
Initial radius $R_1=3R$
Final radius $R_2=4R$
As we know that the energy of satellite is given by

$E=-\frac{G{M}_{e}m}{2r}$

Here, the initial and final energies are given as:

$E_i=-\frac{G{M}_{e}m}{2\left(3R\right)}$  

$E_f=-\frac{G{M}_{e}m}{2\left(4R\right)}$

So, the extra energy required is:

$\Delta E=E_f-E_i=\frac{G{M}_{e}m}{24R}=\frac{mgR}{24}=\frac{1000\times 9.8\times 6400\times {10}^3}{24}$

$\Delta E=2.614\times {10}^9\text{J}$

Hence , option $\left(A\right)$ is correct.