Derivation of Escape Velocity

You might be thinking of how difficult it is to launch a massive object into space. How much velocity does it require to complete this phenomenon? However, all objects, be it a rocket or a baseball moving deep into space, require the same speed. Notably, this velocity or acceleration is known as escape velocity.

What Is Escape Velocity?

Escape velocity in Physics is the speed that an object requires to escape from the gravitational force of the Earth. However, it must not accelerate further.

Also, the escape speed is dependent on several factors. It is determined by scientists that escape rate of an enormous body like a star, or a planet is evaluated using the following escape velocity equation:

Ve = √2GM / R

The expression for escape velocity is derivable by taking initial kinetic energy of a body and initial gravitational potential energy at a certain height.

  • Initial potential energy

PEi = -GMm / Ri

Where, PEi: initial gravitational potential energy in kg-km2/s2

             G: Universal Gravitational Constant = 6.674*10−20 km3/kg-s2

             m: mass of the attracting body (kg)

             M: mass of the escaping body (kg)

             Note: M is much greater than m (M >> m)

             Ri: initial separation between the centres of the bodies (km)

  • Initial kinetic energy

KEi = mve2 / 2

Where, KEi: initial kinetic energy (kg-km2/s2)

            M: mass of the object (kg)

             ve: initial velocity—and thus “escape velocity” (km/s)

Hence, the sum of kinetic and potential energies equals to total initial energy:

TEi = KEi + PEi

TEi = mve2/2 − GMm/Ri

Final energy

Also, gravitational fields are assumed to reach infinity. Hence, a body will move to infinity if initial speed is strong enough. As a result, it “escapes” gravitational pull.

  • Potential energy at infinity

PE = −GMm/R

Where, PE: gravitational potential energy at infinity

             R: infinite separation between the centres of the objects

             As, R = ∞ (infinity), thus PE = 0

  • Kinetic energy at infinity

KE = mv2/2

Where, KE: final kinetic energy

             v: final velocity at infinity

However, at infinity, the velocity of the body is zero: v = 0

Therefore, KE = 0

  • Whole final energy

As, kinetic energy is acting upward and potential energy is moving downward, total energy at the initial location:

TE = KE + PE

Therefore, TE = 0 + 0

  • Escape Velocity Derivation

According to the Law of Conservation of Energy, this total energy remains constant for a closed system. So, in this scenario, the closed system contains two bodies with a gravitational pull between each other. Moreover, no outside force or energy is acting on any of the masses.

Hence, TEi = TE

              KEi + PEi = 0

By substituting values:

            mve2/2 − GMm/Ri = 0

By adding GMm/Ri to both sides:

Therefore, mve2/2 = GMm/Ri

Now for ve2 solution:

            ve2 = 2GM/Ri

By applying square root on both sides:

            ve = ± √(2GM/Ri)

However, in this case gravitational convention is away from the other body, so escape velocity expression is negative:

            ve = − √(2GM/Ri)

Note: Although with regards to this convention, the negative version is right; but in maximum escape velocity books, it is considered to be positive.

Do It Yourself

(i) Gas escapes from the surface of a planet as it acquires an escape speed. So, determine which factors will escape velocity depend on:

1. Mass of planet 2. Mass of the element escaping 3. Radius of planet 4. Temperature of planet

(ii) Choose the correction combination of answers from the following options:

(a) 1 and 2 are correct (b) 1 and 3 are correct (c) 1,2 and 3 are correct (d) 1 and 4 are correct

Along with the above escape velocity formula derivation, there are various related topics which are crucial for your Physics curricula. Consequently, you can download our Vedantu app to easily access study material on these topics, along with accessing online classes from our esteemed faculty.

FAQ (Frequently Asked Questions)

1. What is escape velocity derivation class 11?

Escape velocity derivation of an entity from a comparatively more significant mass is attained by comparing kinetic and potential energy values at a certain point. Moreover, this is done with the application of the Law of Conservation of Energy.

2. What is minimum value of escape velocity?

Earth’s escape velocity is nearest to 11.186km/s. Hence, to escape, a body must have an initial speed of 11.186 km/s.

3. Is escape velocity dependent on mass?

Escape velocity is only dependent on size and mass of a body from which anything is making an effort to escape.