Escape Velocity and Orbital velocity

Did we ever wonder how satellites function after they go into space? At what speeds do they revolve around the Earth? Well, these questions can be answered with these two concepts. After the satellite goes into space, it needs a specific speed to revolve around the Earth in a fixed orbit. With the help of orbital velocity, it does so. If the speed increases, then the satellite may go out of orbit and may fail. Therefore, the minimum speed, which is required to do so, is the escape velocity.

Escape velocity

In kinematics, the initial velocity of a projectile is given by Rmax = u2 / 2g, which means, for a certain initial velocity, the particle can fly away due to the gravitational pull of the Earth.

The minimum amount of velocity for which the particle escapes the spherical gravitational influence of a planet is known as escape velocity. (ve). Theoretically, if a body is provided sufficient escape velocity it may go to infinity.

According to the law of conservation of energy the gravitational force is a conservative force. According to the law of conservation of energy, the minimum amount of velocity is given by:
Ui + Ki = Uf + Kf

When the particle reaches an infinite position, there final potential energy becomes zero after reaching a maximum height. So we can deduce the above relation as:
Ui + Ki = 0. Also, we know,
Ui = -GMm / R, Ki = ½ m ve2
Putting these in the equation, we get,
-GMm / R + ½ m ve2 = 0
-GMm / R = ½ m ve2
Therefore, we get, ve = 2GMR

From the above formula, we can see the escape velocity of a satellite does not depend upon the mass of the satellite. In other words, whatever be the mass of the satellite, escape velocity will always be the same.
The escape velocity required for the satellite to move away from Earth is 11.2 km/s.
If v = ve, the body will escape the spherical gravitational influence. If 0 ≤ v < ve, the body will fall back on Earth or will continue to revolve around the orbit.

Orbital velocity

The velocity required to move the mass around the orbit of the Earth is known as orbital velocity. When the test mass is moving around the main mass in a circular path of radius r, then due to centripetal force (in this case, the gravitational force) tends to attract the test mass towards the main mass. Then the orbital velocity can be expressed in terms of:

Relation between escape and orbital velocities

This shows, escape velocity is 2 times the orbital velocity.
1. When both the escape and orbital velocities are the same, the object will move constantly in the orbit with the same elevation.
2. If the escape velocity is less than orbital velocity, the test mass will collapse and will fall on the planet.
3. If the escape velocity is more than the orbital velocity, the test mass will be free and will float in space.

Some formulae related to the motion of satellite

1. The speed of satellite to revolve around the Earth in the fixed orbit is given by: vo = GMr
2. The ratio of the total distance travelled by the satellite to the orbital velocity is known as the time period of a satellite, and is given by:

, which is the Kepler’s 3rd law.
3. Kinetic energy of a satellite is given by: ½ m r2 ω2
ω = 2π / T
Therefore, K = ½ m r2 (2π / T)2
Substituting the value of T from Kepler’s 3rd law, we have,
K = GMm / 2r
4. Potential energy of a satellite is given by: U = -Gm / r
5. Total energy of a satellite = K + U = GMm / 2r + -Gm / r
E = -GMm / 2r
6. Angular momentum of a satellite is given by:

Difference between escape and orbital velocities

Orbital velocity	Escape velocity
It is the velocity needed for a test mass to revolve around a source mass.	It is the minimum velocity needed for a test mass to move away from its orbit.
It is measured in m/s.	It is measured in m/s.
It is given by: v_o = GMr	It is given by: v_e = 2GMR
If the orbital is less than the escape velocity, the object will float in space.	If the escape is less than the orbital velocity, the object will crash into the planet.