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(C) The total energy of the body at the earth's surface is the sum of its potential energy and kinetic energy.Potential energy at the surface is $U =

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$mgR$

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(C) The total energy of the body at the earth's surface is the sum of its potential energy and kinetic energy.Potential energy at the surface is $U = -\frac{GM_Em}{R}$.To reach infinity,the final total energy must be at least $0$ (where potential energy and kinetic energy are both $0$).By the law of conservation of energy: $U + K.E. = 0$.$-\frac{GM_Em}{R} + K.E. = 0$.$K.E. = \frac{GM_Em}{R}$.Since the acceleration due to gravity at the surface is $g = \frac{GM_E}{R^2}$,we can write $GM_E = gR^2$.Substituting this into the expression for kinetic energy: $K.E. = \frac{(gR^2)m}{R} = mgR$.

The kinetic energy needed to project a body of mass $m$ from the earth's surface (radius $R$) to infinity is

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