Define: Escape velocity and derive mathematically.

Explanation

Then velocity required to overcome the Earth's gravitational force to move into the space is called escape velocity.
On surface of Earth
A, K.E. `=1/2 mv_(esc)^(2)`
B, P.E. `=(Gmm)/R`
C T.E`=P.E.+K.E`
`E_(1)=1/2mv_(esc)^(2)-(GMm)/R`
At infinity
`A, K.E.=0`
`B,P.E.=-(GMm)/(oo)=0:'R+h=oo`
`C,TE.=P.E.+K.E.`
`=0+0=0`
According to the law of conservation of energy
`E_(1)=E_(2)`
`1/2mv_(esc)^(2)-(GMm)/R=0`
`1/2 cancel(m)v_(esc)^(2)=(GMcancen(m))/R`
`v_(esc)^(2)=(2GM)/R`
`V_(esc)=sqrt((2GMxxR)/(R^(2)))`..........[`g=(GM)/(R^(2))`]
`v_(ec)=sqrt(2gR)`
For Earth `v_(esc)=11.2km//s`.