Define escape velocity. Derive an expression for escape velocity. Find the value of escape velocity.

Escape velocity is the minimum velocity with which a body must be projected vertically upward so that it may just escape the surface of the Earth. 
Expression for escape velocity: 
Let a body of mass m be escaped from the gravitational field of the earth.
During the course of motion, let at any instant, body be at a distance r from the centre of the earth. 
                         
The gravitational force between body and the earth is,
                          
Work done to raise the body by distance dr is, 
                      
Total work done, W in raising the body from the surface of the earth to infinity is, 
                              
                          
If we throw the body upward with a velocity v_e, then work done to raise the body from surface of the earth to infinity is done by kinetic energy.
Therefore, 
                    
                        
                                      ... (1) 
Substituting the values in equation (1), we have
g = 9.81m/sec and R = 6.4 x 10⁶m
Escape velocity is given by,