Escape Velocity

Concept of Escape velocity

When standing on Earth, if we throw any object up, it comes back to us. Have you ever wondered why it does come back to us and doesn’t leave the surface of the earth? The possible reason is the backward pull exerting on it. The backward pull exerted by the surface of the earth is termed a gravity. The follow up question would be, if earth pulls every object thrown upward, in backward direction, then why do rockets fly in outer space? The possible reason is, the rocket must have exerted more force than the gravitational pull of the earth, but, what makes this force come into play? Considering the mass of the body is fixed, the force is then dependent on the velocity with which the object is thrown. This means, the velocity should increase and should be beyond a certain limit. The limit of the velocity beyond which the object escapes from the gravitational attraction of the earth is called escape velocity. Similarly, for planets and other celestial bodies, the minimum velocity which they possess, which doesn’t allow them to interact with the gravitational pull of other celestial bodies, is their escape velocity. At escape velocity, the sum total of gravitational potential energy and kinetic energy of the object into consideration will be zero. In other words when an object leave’s the surface of a planet or moon, it leaves without any propulsions and development.

Mathematical Expression of Escape velocity

Mathematically escape velocity can be obtained by equating the kinetic energy and potential energy of the object of mass m moving with velocity v, it can be presented by:

vc=2GMr

vc denotes escape velocity of the object

G represents universal gravitational constant

M denotes the mass of the object which is exerting gravitational pull on the object thrown.

r denotes the distance of the object from the centre of the object exerting the gravitational pull.

From the mathematical expression, it is clear that larger the mass of the bodies exerting gravitational pull, greater will be the escape velocity.

For example, earth have radius 6.38 × 106 m and the mass 5.98 × 1024 kg, the escape velocity can be calculated as:

vc=2GMr 

vc= 2×6.673×10-11×5.98×1024×6.38×106 

vc= 11184 m/s = 11.2 km/s

This clearly means that, when any spacecraft is launched into outer space from the earth, it should leave with a velocity greater than 11.2 km/s in order to outshine the gravitational pull of the earth. If the spacecraft is launched with a velocity less than 11.2 km/s, it will be pulled back to the earth’s surface. Also, the escape velocity varies at points where there are visible variations in the radius of the planet. For example, the shape of the planet earth is ellipsoid, i.e. the equatorial radius is different from polar radius. Hence, the escape velocity value will be different at earth’s poles and equator.

Let us explore another example, to calculate the escape velocity of the moon of the earth. The radius and mass of the moon are 1.74 × 106 m, and 7.35 × 1022 kg respectively.

Consider the mathematical expression to calculate escape velocity:

vc=2GMr 

vc= 2×6.673×10-11×7.35×1022×1.74×106 

vc= 2374 m/s = 2.37 km/s

This means, in order to escape from the surface of the moon, the object should leave the surface with a velocity greater than 2.37 km/s. If the object leaves the moon’s surface with velocity less than 2.37 km/s, the object will fall back under the moon’s gravity.

Escape velocity for the black hole is infinite. The gravitational field of the black hole is so high that the value of escape velocity is more than the speed of the light. Hence, it is impossible to escape from the surface of a black hole.

Conclusion

In astronomy and space research, escape velocity is the velocity required for a body to escape from a gravitational centre of attraction without additional acceleration. The velocity required to maintain a circular orbit at the same height decreases with altitude and is equal to the square root of 2 (or around 1.414) times the velocity required to escape. If air resistance was ignored at the Earth’s surface, escape velocity would be around 11.2 kilometres (6.96 miles) per second. At its surface, the velocity of escape from the less massive Moon is around 2.4 kilometres per second. A planet (or satellite) cannot sustain an atmosphere for long if its escape velocity is close to the average velocity of the gas molecules that make up the atmosphere.