Define escape velocty. Obtain an expression for the escape velocity of a body from the surface of earth.

Explanation

Escape velocity `(v_(e))`. It is minimum velocity given to a body , so that it goes out of the gravitational pull of the earth. Consider a body of mass m at a distance r from the centre of earth. The gravitational force acting on it is given by
`F= (GMm)/(r^(2))` ...(i)
Then, work done to displace the body by distance dr is given by
`dW= Fdr = (GMm)/(r^(2)) dr` [using equation (i)]
Now total work done to displace the body from the surface of earth to infinity is given by
`W= int dW= underset(r)overset(oo)int (GMm)/(r^(2)) dr= GM m underset(R )overset(oo)int (1)/(r^(2)) dr`
or `W= GM m [-(1)/(r )]_(n)^(oo) = - GM m[(1)/(oo)- (1)/(R )]`
or `W= (GMm)/(R )`

The body will escape from gravitational pull, if kinetic energy of body is equal to total work done i.e., `(1)/(2) mv_(e)^(2) = (GM m)/(R )`
or `v_(e)= [(2GM)/(R )]^(½)` ...(ii)
or `v_(e) = sqrt(2gR) (because g= (GM)/(R^(2)))` ...(iii)
Putting `g= 9.8 ms^(-2), R= 6.4 xx 10^(6)m`, we get
`v_(e)= sqrt(2 xx 9.8 xx 6.4 xx 10^(6))= 11.2 km s^(-1)`