It is the minimum speed with which a body must be projected from the surface of a planet (usually the earth) so that it permanently overcomes and escapes from the gravitational field of the planet (the earth). We can also say that a body projected with escape speed will be able to go to a point which is at infinite distance from the earth.

If a body of mass `m` is projected with speed `v` from the surface of a planet of mass `M` and radius `R`, then

`K.E.=1/2mv^2` ; `G.P.E.=-(GMm)/R`

Total mechanical energy (`T.M.E.`) of the body

`=1/2 mv^2-(GMm)/R`

If the `v` is the speed of the body at infinity, then `T.M.E.` at infinity `=0+1/2 mv'^2=1/2mv'^2`

Applying the principle of conservation of mechanical energy.

we have

`1/2mv^2-(GMm)/R = 1/2 mv'^2` or, `v^2=(2GM)/R+v'^2`

v will be minimum when `v'-> 0,` i.e.,

`v_e=v_(min)=sqrt((2GM)/R)=sqrt(2gR)`