Question

What is orbital velocity? Obtain an expression for it.

Answer 1

Hint: Orbital velocity, velocity sufficient to cause a natural or artificial satellite to remain in orbit. Inertia of the moving body tends to make it move on in a straight line, while gravitational force tends to pull it down. For this derivation, we will assume a uniform circular motion. Therefore, for an object in orbit, both these forces will be equal.

Complete step-by-step solution:
Orbital velocity refers to the velocity required by satellites {natural or artificial) to remain in their orbits. The orbit or the orbital pathway, whether elliptical or circular, displays a balance between the inertia of the satellite which makes it move in a straight line with the gravity of the planet which pulls the satellite closer to the planet. Now, this orbital velocity depends on the distance between the center of the planet and the satellite revolving around it.
Formula for orbital velocity:
The orbital velocity equation is given by:
$v = \sqrt {\dfrac{{GM}}{r}} $
Where,
$R$ is the radius of the orbit,
$G$ is the gravitational constant and
$M$ is the mass of the central body of attraction.
Orbital velocity formula derivation-
The following steps can be followed to derive an expression for the orbital velocity of a satellite revolving in an orbit. For orbital speed derivation, both the gravitational force and centripetal force are very important. Gravitational force ${F_g}$ is the force exerted by the body at the center to keep the satellite in its orbit. Centripetal force ${F_c}$ justifies the existence of circular motion.
For this derivation, we will assume a uniform circular motion. Therefore, for an object in orbit, both these forces will be equal.
$ \Rightarrow {F_g} = {F_c} - - - - - (1)$
Now, as we know gravitational force depends on the masses of both objects and its formula is:
${F_g} = \dfrac{{GMm}}{{{r^2}}}$
$G$ is the gravitational constant, $M$ is the mass of the central body, and $m$ is the mass of the revolving body, and $r$ is the distance between the two bodies.
Also, the centripetal force is related to mass and acceleration, and the formula for which is given by:
${F_c} = \dfrac{{m{v^2}}}{r}$
Where,
$m$ is the mass of the satellite in orbit, $v$ is the speed, $r$ is the radius of the orbit.
Now, putting both of these equations in (1), we get,
\[
  {F_g} = {F_c} \\
   \Rightarrow \dfrac{{m{v^2}}}{r} \\
   \Rightarrow \dfrac{{GM}}{r} = {v^2} \\
   \Rightarrow v = \pm \sqrt {\dfrac{{GM}}{r}} \\
 \]
Therefore, orbital velocity derivation is proved.

Note: There is difference between escape velocity and orbital velocity escape velocity is the base speed required for a free, non-propelled object to escape from the gravitational impact of a huge center body, that is, to move an unlimited distance from it. This velocity is a component of the mass of the body and separation to the focal point of mass of the body.