Escape Velocity and Orbital Velocity

Relation Between Escape Velocity and Orbital Velocity

The lowest velocity an object must have to escape the gravitational force of a planet or an object. The relationship between the escape velocity and the orbital velocity is defined by Ve = 2 Vo where Ve is the escape velocity and Vo is the orbital velocity. And the escape velocity is root-two times the orbit velocity.

Escape velocity, as it relates to rocket science and space travel, is the velocity required for an object (such as a rocket) to escape the gravitational orbit of a celestial body (such as a planet or a star).

We have studied in kinematics that the range of the projectile depends on the initial velocity of the projectile. ⇒ Rmax∝u2 ⇒ Rmax= u2/2g, which means that the particle flies away from the gravitational impact of the earth at a certain initial velocity provided to the particle.

This minimum amount of velocity for which the particle escapes the gravitational sphere of influence of the planet is known as the velocity of escape (ve). When an escape velocity is given to a body, it theoretically goes to infinity.

As gravitational force is a conservative force, the law on energy conservation is fine. Applying the law on the conservation of energy for a particle with the necessary minimum velocity to infinity

Ui + K= Uf + Kf

At infinity, the particles undergo no interaction, so the final potential energy, and we know from motion in the 1D chapter that the final velocity of the body is zero after reaching its maximum height so that we can deduce the final kinetic energy of the particle.

Then ,

Ui + K= 0 and   we   know   that, Ui ​= −GMm​ / R , Ki​= 1/2​mve2

We get,

1/2​mve2 + ( −GMm/ R ​) = 0 ⇒ 1/2​ mve2 = GMm​ / R

That implies,

ve​= √ 2GM / R ​​ ……………(1)

It is obvious from the above formula that the escape velocity does not depend on the test mass (m). If the source mass is earth, the escape velocity has a value of 11.2 km / s. When v = ve the body leaves the gravitational field or control of the planets, when 0 ≤v < ve the body either falls down to Earth or proceeds to orbit the earth within the sphere of influence of the earth.

Orbital Velocity

Orbital velocity is the velocity that the body will sustain in order to orbit another body. Escape velocity is the speed at which an object leaves the orbit. Escape velocity will be a square-root of 2 times the orbital velocity in order to exit the orbit.

(l) If the velocity is equal, the body must remain in constant orbit, not in elevation.

(ll) less than the orbit, the orbit will decay and the object will crash.

(ll) rather than orbital, and the body will have an ascending orbit, which will fly out into space.

The speed at which the test mass travels around the source mass is known as orbital velocity

(vo) when the test mass orbits around the source mass in a circular path of radius 'r' having the center of the source mass as the center of the circular path, the centripetal force is provided by the gravitational force as it is always the attracting force having its direction towards the center of a source mass.

⇒ mvo​2​  / r = GMm​ \ r2

⇒ vo​2​/r = GM​​ \ r2

⇒ vo ​= √GM​​ /r

If the test mass is small distances from the source mass  r ≈ R (radius of the source mass)

Then,

vo​ = √GM​​ /r …………..(2)

The above formula indicates that the orbital velocity is independent of the test mass (the mass which is orbiting).

Relation Escape Velocity And Orbital Velocity Formula

In astrophysics, the relationship between escape velocity and orbit velocity can be mathematically described as -

Vo=Ve2√

Or

Ve=2–√Vo

Where,

  • Ve is the Escape Speed Measurement using km / s.

  • Vo is the calculation of angular velocity using km / s.

We know that Escape Velocity=2x√Orbital Velocity that means that the escape velocity is directly proportional to the orbital velocity. This means for any big body-

• When the angular velocity increases, the escape velocity will also increase and aim-versely.

• If the angular velocity decreases, the escape velocity decreases as well as the vise-versa.

Escape Velocity and Orbital Velocity

The relationship between escape velocity and orbital velocity equations is very important for understanding the definition. For any kind of massive body or planet.

  • Escape velocity is given by – Ve=2gR−−−−√ ———-(1)

  • Orbital velocity is given by – Vo=gR−−−√ ———–(2)

Where,

g is the acceleration due to gravity.

R is the radius of the planet.

From equation (1) we can write that-

Ve=2–√gR−−−√

Substituting Vo=gR−−−√ we get-

Ve=2–√Vo

The above equation can be rearranged for orbital velocity as-

Vo=Ve2√

FAQ (Frequently Asked Questions)

Q1. How Does Escape Velocity Work?

Ans - Escape velocity is the minimum velocity required to transcend the gravitational force of a large body and escape into infinity. Orbital velocity is the velocity at which the object revolves around a massive body. The relationship between the escape velocity and the orbital velocity is proportional.

Like orbital velocity, the escape velocity varies depending on the distance the object is from the center of gravity. In practical terms, the higher the rocket's altitude is above Earth, the lower the velocity:

  • Orbit the Earth

  • Escape Earth’s gravitational field altogether

One reason that communication satellites can orbit the Earth without constantly consuming energy is that they subsist at an altitude of miles above the Earth. On the other hand, commercial aircraft, which travel far closer to the surface of the earth, will continuously use energy to stay in the atmosphere. 

Q2. Explain Orbital Velocity?

Orbital velocity is the speed required to reach orbit around a celestial body, such as a planet or a star, while escape velocity is the speed required to leave that orbit. Maintaining orbital velocity requires travel at a sustained speed that:

  • Align with the rotational velocity of the celestial body

  • Is fast enough to counteract the force of gravity pulling the orbiting object towards the surface of the body.