if velocity given to an object from the surface of the earth is n times the escapes velocity then what will be the residual velocity at inifinity.

Explanation

Let the residual velocity be v, then from energy conservation `(1)/(2)m(nv_(e))^(2)-("GMm")/( R )`
`=(1)/(2)mv^(2)+0`
`impliesv^(2)=n^(2)v_(e)^(2)-(2"GM")/( R )=n^(2)v_(e)^(2)=(n^(2)-1)v_(e)^(2)`
`impliesv=(sqrt(n^(2)-1))v_(e)`.