Answer
Using Theorem 2, we show that the angular velocity $\theta '\left( t \right)$ is constant. Hence, the planet in a circular orbit travels at constant speed.
Work Step by Step
We have ${\bf{J}} = {\bf{r}}\left( t \right) \times {\bf{r}}'\left( t \right)$. By Theorem 1, ${\bf{J}}$ is constant. So, $J = ||{\bf{J}}||$ is constant.
By Theorem 2, we have $J = r{\left( t \right)^2}\theta '\left( t \right)$. Since the orbit of the planet is circular, the radius of the orbit $r\left( t \right)$ is constant. Therefore, the angular velocity $\theta '\left( t \right)$ is also constant. Hence, the planet in a circular orbit travels at constant speed.
