Question

What is meant by escape velocity? Write the relation between escape velocity and orbital velocity.

Answer 1

Hint-As we must have seen rockets leaving the earth to go into space. Then you must have also noticed they require a very huge kick-start in order to leave the surface of the earth. It is because of the strong gravitational field of the surface of the earth. Thus, this is where escape velocity comes in. while Orbital velocity is the velocity with which an object revolves around a massive body.

Complete step-by-step answer:

Escape velocity refers to the minimum velocity which is needed to leave a planet or moon. For instance, for any rocket or some other object to leave a planet, it has to overcome the pull of gravity.
Orbital velocity is the velocity at which a body revolves around the other body. Objects that travel in the uniform circular motion around the Earth are called to be in orbit. The velocity of this orbit depends on the distance between the object and the centre of the earth.
We know that Escape velocity = $\sqrt{\text{2}}$$\times$ orbital velocity which implies, the escape velocity is directly proportional to orbital velocity. That means for any massive body-
 ${\text{V}}_{\text{e}}\text{ = }\sqrt{\text{2}}{\text{V}}_{\text{O}}$
Or ${\text{V}}_{\text{O}}\text{ = }{\displaystyle \frac{{\text{V}}_{\text{e}}}{\sqrt{\text{2}}}}$
Where, ${\text{V}}_{\text{e}}$ is the Escape velocity measure using km/s.
              ${\text{V}}_{\text{O}}$ is the Orbital velocity measures using km/s.
Derivation of relation between escape velocity and orbital velocity: -
Escape velocity is given by-
${\text{V}}_{\text{e}}$= $\sqrt{\text{2gR}}$ ……………. (1)
Orbital velocity is given by-
${\text{V}}_{\text{o}}$= $\sqrt{\text{gR}}$ ……………… (2)
Where,
g is the acceleration due to gravity.
R is the radius of the planet.
Now, from (1) and (2), we get
  ${\text{V}}_{\text{e}}\text{ = }\sqrt{\text{2}}{\text{V}}_{\text{O}}$
The above equation can be rearranged for orbital velocity as-
${\text{V}}_{\text{O}}\text{ = }{\displaystyle \frac{{\text{V}}_{\text{e}}}{\sqrt{\text{2}}}}$

Note- Therefore, we understood the relation between escape velocity and orbital velocity for any object for any massive body or planet. If orbital velocity increases, the escape velocity will also increase,iIf orbital velocity decreases, the escape velocity will also decrease .