An object is thrown from Earth is such a way that it reaches a point at infinity with non-zero kinetic energy `[K.E(r=infty)=(1)/(2)MV_(infty)^(0)]`, with what velocity should the object be thrown from Earth?

Explanation

From the law of conversation of energy
`E_(i)=E_(f)`
`(1)/(2)mV_(e)^(2)-(GM_m)/(R_E)=(1)/(2)mv^(2)-(GM_m)/((R+h))`
At `h=infty,V=V_(infty)`, For non-zero K.E.
`(1)/(2)mV_(e)^(2)-(GM_m)/(R_E)=(1)/(2)mv_(infty)^(2)`
`(1)/(2)mV_(e)^(2)=(1)/(2)mv_(infty)^(2)+(GM_m)/(R_E)`
`(1)/(2)mV_(e)^(2)=(1)/(2)m(v_(infty)^(2)+(2GM)/(R_E))`
`V_(e)^(2)=v_(infty)^(2)+(2GM)/(R_E)(becauseg=(GM)/(R_(E)^(2)))`
`V_(e)^(2)=v_(infty)^(2)+(2gR_(E)^(2))/(R_E)`
`V_(e)=sqrt(v_(infty)^(2)+2gR_(E))`