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`
`implies v^(2)=n^(2) v_(e)^(2) - (2GM)/R= n^(2) v_(e)^(2)-v_(e)^(2)= (n^(2)-1) v_(e)^(2) implies v= (sqrt(n^(2)-1)) v_(e)`.