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Escape Speed

If we throw a ball upwards, it reaches a certain height and then falls back due to the attraction of the earth. If it is thrown with a greater initial velocity, it rises to a greater height.
We shall show just now that if, somehow, the body is projected with a velocity of about 11.2 km s-1, it will break loose from the earth and never come back.
In other words, it will escape the gravitational pull of the earth and go into outer space. The escape velocity is the minimum velocity with which a projectile must be projected in order that it may escape the earth's gravitational pull. The magnitude of the escape velocity can be deduced as follows.
Consider a body of mass m at a distance r from the centre of the earth of mass M. clearly, the gravitational force acting on the body is
Therefore, the work 'dw' done by the body against the gravitational pull of the earth in moving upward through a further small distance 'dr' is given by

Hence, the total work W done by the body in escaping from the surface of the earth, i.e. in moving to an infinite distance away form it, is given by
w =
Where R is the radius of the earth.
Let υe be the escape velocity. Then the initial kinetic energy of the body is . This must at least be equal to the work done by the body in escaping from the earth. We, therefore, have
The expression can also be written in terms of g, the acceleration due to gravity at the earth's surface.
Putting g = 9.8 ms-2 and m, we have
υe = 11.2 km s-1
For planet Mercury, the escape velocity is about 4.2 km s-1, for Jupiter it is 61 km s-1 and for Moon the escape velocity is 2.3 km s-1.

An Interesting Consequence of Escape Velocity

The escape velocity is independent of the mass of the body. Thus, very massive rockets and extremely tiny particles (such as the molecules of a gas) will require the same initial velocity to escape from the earth or from any other planet or star.
We shall learn that the molecules of gas move around with a certain average velocity that depends on the nature and temperature of the gas.
For oxygen, nitrogen, carbon dioxide and water vapour, the average velocity at moderate temperatures is of the order of (0.5 to 1.0) 103 ms-1 or (0.5 to 1.0) km s-1, whereas for lighter gases, such as hydrogen and helium it is (2 to 3) km s-1 .
Now, we know that the moon has no atmosphere. The velocity of escape from the moon is about 2.5 km s-1. It is clear that lighter gases, such as hydrogen and helium, whose average molecular velocities are of the order of the escape velocity, will escape from the moon. 
The gravitational pull of the moon is too weak to hold these gases. Thus, in course of time, the gases that were supposed to be present in the atmosphere of the moon, have escaped from it.
The presence of the lighter gases in relative abundance in the atmosphere of the sun should not surprise us in view of the much stronger gravitational attraction of the sun and consequently much higher escape velocity which, for the sun, is about 620 km s-1.

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