Question

derivation of escape velocity

Answer 1

Suppose that the planet be a perfect sphere of radius R having mass M. Let a body of mass m is to be projected from point A on the earth's surface as shown in the figure. Join OAand produce it further. Let us take two points P and Q which are at distances x and (x +dx) from the center of the earth.

To calculate the escape velocity of the earth, let the minimum velocity to escape from the earth's surface be ve. Then, kinetic energy of the object of mass m is

When the projected object is at point P which is at a distance x from the center of the earth, the force of gravity between the object and earth is

Work done in taking the body against gravitational attraction from P to Q is given by

The total amount of work done in taking the body against gravitational attraction from surface of the earth to infinity can be calculated by integrating the above equation within the limits x = R to x = ∞. Hence, total work done is

For the object to escape from the earth's surface, kinetic energy given must be equal to the work done against gravity going from the earth's surface to infinity, hence

K.E. of Object should be Equal to Magnitude of P.E.

Since,

The relation shows that the escape velocity of an object does not depend on the mass of the projected object but only on the mass and radius of the planet from which it is projected

Answer 2

we can derive the equation for the gravitational escape velocity by considering the initial gravitational potential energy at the given altitude and the initial kinetic energy of the object. By comparing this total initial energy with the sum of the potential and kinetic energies at an infinite separation, you can determine the escape velocity equation.