Answer to Question #132824 in Mechanics | Relativity for max

Question

explain the relationship between escape velocity and orbital velocity.

Answer 1

Solution. The concept of the orbital velocity is very simple - it is the speed that must be given to a physical object so that it, moving parallel to the cosmic body, could not fall on it, but at the same time it would remain in a constant orbit. Equating the centripetal force with the force of gravitational attraction, we obtain the formula for orbital velocity

$$v_{orb}=\sqrt{\frac{GM}{R}}$$

where G=6.67x10^-11 N kg^2/m^2 is gravitational constant; M is planet mass; R is orbit radius.

The second cosmic speed is the minimum speed with which a body must move so that it can overcome the influence of the Earth's gravitational field without spending additional work, i.e. move an infinitely large distance from the planet.

In this case, the kinetic energy of the body should be equal to the work to overcome the influence of the gravitational field:

$$\frac {mv_{esc}^2}{2}=\frac{GmM}{R}$$

where m is body mass; G=6.67x10^-11 N kg^2/m^2 is gravitational constant; M is planet mass; R is distance from planet to body.

As result

$$v_{esc}=\sqrt{\frac{2GM}{R}}$$

Therefore the relationship between escape velocity and orbital velocity

$$v_{esc}=\sqrt{2}\times v_{orb}$$