Relationship between Escape and Orbital Velocity

The simplest way to understand the relationship between escape and orbital velocity is to learn how rockets take off. A push helps the rocket leave the Earth’s surface and enter the space circuit. The Earth’s gravitational field is powerful, and the objects need a hard ‘push’ to move against it and enter space.

Escape velocity refers to any object’s minimum force required to leave the Earth, going against the gravitational forces. This velocity helps in successful escalation from the planet despite the solid gravitational pull.

On the other hand, orbital velocity is the force or pressure required for any object or body to rotate around the Earth’s orbit. Therefore, it is essential to understand their significance for determining the motion of satellites around the planet or any other objects that travel from Earth to space or vice-versa.

Escape Velocity: What is it and Formula?

Gravitational field around Earth’s surface attracts every object downwards after moving in the opposite upward direction. This is due to the gravitational force of the planet. If any object needs to disperse this field and move to space, it requires a certain velocity. This velocity is termed as the ‘Escape Velocity’. With a certain amount of push, the object would surpass the influence of gravity (E.V.), and then it won’t get back to the ground.

Escape velocity is used to determine space travel time and rocket science. The kinematics state that the projectile range varies as per its initial velocity. 

Hence,

Rmax ∝ u2

Rmax = u2/2g

This indicates that when a projectile is given a particular velocity of push, it flies through the gravitation effect of the planet and releases itself into infinity.

The energy conservation law is also applicable here as the gravitational force is a conservative one.

Ui + Ki = Uf + Kf

Once the object flies to infinity, its potential velocity reaches zero at the maximum height. 

Here, we derive the kinetic energy of the object through this Formula:

Ui + Ki = 0

You know that Ui = -GMm/R

and Ki = ½ mve2

 thus, we get

½ mve2 + (-GMm/R) = 0

½ mve2 = GMm/R

Hence, we get 

ve = √2GM/R

The escape velocity remains stagnant despite the mass of any projectile as it is irrelevant to that. 

Orbital Velocity: What is it and Formula?

Orbital velocity is the force required by an object like (Satellite) to stay in the orbit of another object like (Earth). You can get the exact value of orbital velocity by square rooting the object’s escape velocity two times. 

· When escape and orbital velocity are identical, zero elevation results in a constant orbit. 

· When the velocity is on the lower side to orbit, it would decay, resulting in a crash of the object. 

If the test mass revolves around the orbit in a circular direction, it has the radius ‘r’. This path has the source mass in the centre as the gravitational force attracts it towards the centre of the source mass. 

Thus, we get

mvo2/r = GMm/r2

vo2/r = GM/r2

vo = √GM/r

when test mass is at a smaller distance which is r ~ R, we get

vo = √GM/r

Relationship Between Escape and Orbital Velocity

Before delving into the relationship between escape and orbital velocity examples, let’s understand its mathematical interpretation. 

vo = ve/√2

ve = √2vo

Here, ve is denoted as the escape velocity measured in km/s

vo is denoted as the orbital velocity measured in km/s.

Thus

Escape velocity = √2 x Orbital velocity

Hence it is proved that the relationship between escape and orbital velocity is directly proportional. 

In simple words, 

· When the orbital velocity increases, the escape velocity rises in the same proportion. 

· When the escape velocity decreases, the orbital velocity also goes down. 

As a result, 

The escape velocity of the object- ve = √2gR —- 1

Orbital velocity of the subject- vo = √gR —- 2

g is here the accelerated force rising due to gravity. 

Equation 1

ve = √2 √gR

after substituting it with vo = √gR, 

we get

ve = √2Vo

The rearranged equation for orbital velocity is 

vo = ve/√2

Now you know the relationship between escape and orbital velocity importance and its Formula. 

Conclusion

Hopefully, now you are well-versed with the relationship between escape and orbital velocity examples, importance, and formulas slated down above. Astronomical experts and scientists use this relationship equation to analyse the motion of objects in the satellite after they are released from the celestial body of the planet earth. 

The crux of this subject is when both the escape and orbital velocity are the same, the body will stay constant in orbit.