Define escape velocity. Show that the square of escape velocity is equal to the product of diameter of the earth and acceleration due to gravity at the earth.

Escape velocity is the minimum velocity with which a body must be projected vertically upward in order that it may just escape the gravitational pull of the earth.
                          
Let M be the mass of the earth, and
R the radius of the earth.
Let at any instant, mass m be at P which is at a distance r from the centre of the earth.
The gravitational force on mass m is, 
                         
The work done by external force to raise the body by small distance dr against gravity is, 
              
Total work done in taking the body from surface of the earth to infinity is, 
           
              
If the body is at surface of the earth and if kinetic energy is  the mass would escape from gravitational pull of the earth.
i.e.,            
                        
                         
That is, escape velocity is the product of diameter of the earth and acceleration due to gravity at the earth. 
Hence the result.