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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.

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Let 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)-(2GM)/(R)=n^(2)v_(e)^(2)-v_(e)^(2)-v_(e)^(2)=(n^(2)-1)v_(e)^(2)impliesv=(sqrt(n^(2)-1))v_(e)`
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