Question

What is the relation between escape and orbital velocity?

Answer 1

1. Escape velocity (speed) is the minimum speed required by a non-accelerating object to overcome the gravitational pull of a massive body (like Earth) and escape into space.
2. The expression for escape velocity is, $v_e=\sqrt{\frac{2GM}{r}}$. Here, $v_e$ is the escape speed, $G$ is the gravitational constant, $M$ is the mass of the massive body and $r$ is the distance between the initial position of the object and the center of the body.
3. Orbital velocity is the speed at which an object is revolving around a massive body along its orbit. A natural or artificial satellite's orbital velocity is the velocity required to keep it in orbit. The expression for the orbital velocity will be, $v_o=\sqrt{\frac{GM}{r}}$.
4. Comparing the expressions for $v_e$ and $v_o$, we can get a relation between them.

$\begin{array}{rcl}{v}_e& =& \sqrt{\frac{2GM}{r}}\\ v_e& =& \sqrt{2}\sqrt{\frac{GM}{r}}\\ v_e& =& \sqrt{2}{v}_o\end{array}$

Hence, the relation between escape and orbital velocity is $v_e=\sqrt{2}{v}_o$.