Derivation of Escape Velocity: Definition, Formula and Derivation

Derivation of Escape Velocity of an entity can be obtained by comparing kinetic and potential energy values at a specific point. Furthermore, it is calculated with the application of the Law of Conservation of Energy which states that energy can neither be created nor destroyed. It can be transformed from one form to another e.g potential energy of a ball at a certain height can be converted to its kinetic energy if the ball falls from a height.

Key Terms: Escape Velocity, Gravitation, Escape Velocity of Earth, Escape Velocity of Moon, Escape Speed

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What is Escape Velocity?

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Escape velocity can be defined as the minimum velocity value that an object should acquire to be launched into space so that it overcomes the gravitational force of earth or any other planet. The conditions that we need to be mindful of for this mechanism are that the object needs to reach an infinite distance from the planet and that the object should be free.

The general formula for escape velocity is:

Escape Velocity

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Important Formulas

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Derivation of escape velocity involves the relationship between an object and its velocity. It also includes kinetic energy and gravitational potential energy. We use the initial kinetic and initial gravitational energy for a particular height to find escape velocity. Let's see how we can find the final formula of the escape velocity.

Initial Kinetic Energy

We know that kinetic energy is defined as the energy of an object in its motion. We are using the kinetic energy value where the initiation of the movement happened. The formula for initial kinetic energy is:

Where,

• KE_i is the initial kinetic energy
• m is mass of the body
• ve is the initial velocity which is taken in km/s and will eventually give us escape velocity

Initial Potential Energy

Potential energy is the amount of energy stored in an object when it is in a stationary state. The formula for initial potential energy is:

Here,

• PE denotes the initial gravitational potential energy which will be taken in kg-km²/s²
• G is a gravitational constant which has a value of 6.674 X 10^-20 km³/kg-s²
• M is the mass of the bigger body which is exerting gravitational force
• m is taken as the mass value for the escaping body
• R is the value for the initial distance between both objects

This distance will be measured from the center of the body in km. 

Remember that M has a larger value than m.

Taking the sum of initial kinetic and potential energy values, we will get:

TE_i = KE_i + PE_i

Now, we will calculate both of these energies at infinity.

Potential Energy at Infinity

The formula for potential energy at infinity is:

Here,

PE_n is the gravitational potential energy at infinity distance

R­_n is the infinite distance between both objects which is taken from the center of the objects.

From the infinite distance, the value of PE_n is 0.

Kinetic Energy at Infinity

The formula for kinetic energy at infinity distance is:

KE­­_n = mv_n²/2

Here, KE_n is the kinetic energy at infinity distance,

m is the mass of the object, and

v_n is the final velocity at infinity distance. 

Now the value of the KE_n is 0 because v_n is also 0.

Whole Final Energy

As the potential energy acts downward and the kinetic energy moves upward, the whole final energy at the initial location from all the above formulas will be:

TE_n = 0 + 0 

= 0

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Derivation of Escape Velocity

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Considering the law of conservation of energy, if we take a closed system then all the energy remains conserved. Let's assume that the closed system encompasses two objects. Both of them are attracting each other through gravitational force. So in that case,

TE_i = TE

KE_i+ PE_i = 0

If we put the values in the above equation then,

mv_e²/2 - GMm/R_i = 0

Now, if we open the v_e²= 2GM/R_i

Applying square root on both sides then, the equation will become

v_e= √(2GM/R_i)

Now as the gravitation convention is away from another body, the velocity expression will be negative-

v_e = –√(2GM/R_i)

As per the convention, the escape velocity is negative. However, the escape velocity is generally taken as positive to get the maximum value.

Also check: Universal Gravitation Formula

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Escape Velocity of Earth

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v_e = √(2GM/R_i)

Putting g = GM

v_e = √(2gR_i)

Hence, the escape velocity of a planet depends on the radius and mass of the planet and not on the mass of the object.

Now, for Earth

g = 9.8 m

R = 6378100 m

Hence, the escape velocity for Earth becomes:

v_e = √2 x 9.8 x 6378100

= 11.2 km/s

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Things to Remember based on Derivation of Escape Velocity

• Escape velocity can be defined as the minimum velocity value that an object should acquire to be launched into space so that it overcomes the gravitational force of earth or any other planet.
• Derivation of escape velocity involves the relationship between an object and its velocity.
• Initial Kinetic Energy: KE_i = mv_e²/2
• Initial Potential Energy: PE_i = -GMm/R_i
• Potential Energy at Infinity: PE_n = -GMm/R_n
• Kinetic Energy at Infinity: KE­­_n = mv_n²/2
• Escape Velocity Formula: v_e = √(2GM/R_i)
• Escape Velocity of Earth: 11.2 km/s

Also check: