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

Escape velocity is minimum velocity with which a body must be thrown upward so that it may just escape.
Expression for escape velocity:
Let a body of mass m be escaped from 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,
                               
and 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,
                                 
or                                
or                                 
Substituting the values, g = 9.81m/sec and R = 6.4 x 10⁶m, we have,
  
       

(i) Escape velocity is independent of the mass of body projected and is same for all the bodies.

i.e., 11.2 km/s, therefore, both must be projected with equal velocity. 

(ii) Momentum of body of mass m₁ is,
                       

Momentum of the body of mass  is, 

                       

Since     

∴              

That is, a heavy body must be thrown with greater momentum. 

(iii) Kinetic energy of mass  is, 

            

Kinetic energy of mass  is,

                 

Since m₁ > m₂, therefore K₁ > K_2.

That is, i.e., heavy body must be thrown with greater kinetic energy. 

Orbital velocity is the velocity given to artificial satellite so that it may start revolving around the earth.

Expression for orbital velocity:
Suppose a satellite of mass m is revolving around the earth in a circular orbit of radius r, at a height h from the surface of the earth. Let M be the mass of the earth and R be radius of the earth.
∴ r = R + h
    To revolve the satellite, centripetal force of  is required (where  is orbital velocity)  which is provided by gravitational force  between the earth and the satellite.
∴               
or             

or               
But GM=gR² where g is the acceleration due to gravity.

Let g' be the acceleration due to gravity in the orbit i.e. at a height h from the surface.
∴       
or            
∴