Question

By what percent the energy of the satellite has to be increased to shift it from an orbit of radius $r$ to $\frac{3r}{2}$

• A. $66.7\%$
• B. $33.3\%$
• C. $15\%$
• D. $20.3\%$

Answer 1

The correct option is B $33.3\%$

Gravitational potential energy at distance $r$

$U_r=\frac{-GMm}{r}$

Gravitational potential energy at distance $\frac{3r}{2}$

$U_{3r/2}=\frac{-GMm}{3r/2}$

$U_{3r/2}=\frac{-2GMm}{3r}$

Change in gravitational energy after shifting from $r$ to $3r/2$

$U=\frac{-2GMm}{3r}-\left(-\frac{GMm}{r}\right)$

$U=\frac{GMm}{r}\left(1-\frac{2}{3}\right)$

$U=\frac{GMm}{3r}$

Percentage change in gravitational potential energy

$=\frac{U}{U_r}\times 100$

$=\frac{GMm}{3r}\times \frac{r}{GMm}\times 100$

$=33.3\%$