Question

Satellites A and B are orbiting around the earth in orbits of ratio $R$ and $4R$ respectively. The ratio of their areal velocities is

• A. $1:2$
• B. $1:4$
• C. $1:8$
• D. $1:16$

Answer 1

The correct option is A $1:2$

Using Kepler's second law we have:
$Arealvelocity=\frac{Areaswept}{time}=\frac{\frac{1}{2}r\left(rd\theta \right)}{dt}=\frac{1}{2}{r}^2\omega =\frac{1}{2}rv=constant$

Thus we get, $\frac{1}{2}\sqrt{\frac{GM}{r}}r$ (from $\frac{m{v}^2}{r}=\frac{GMm}{r^2}$)
Thus we get
Areal velocity $\propto \sqrt{r}$
Thus for the planets with radius in ratio of $1:4$ we get radii in the ratio of $1:2$