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To project a body of mass $$m$$ from earths surface to infinity, the required kinetic energy is (assume, the radius of earth is $$R_E$$, $$g =$$ acceleration due to gravity on the surface of earth):
To just reach infinity with zero final speed, total mechanical energy at infinity should be zero.
At Earth’s surface:
Potential energy is
$$U=-\frac{GMm}{R_E}$$
Required kinetic energy = escape energy so that
$$K+U=0$$
Thus
$$K=\frac{GMm}{R_E}$$
Now use
$$g=\frac{GM}{R_E^2}$$
So
$$GM=gR_E^2$$
Substitute:
$$K=\frac{(gR_E^2)m}{R_E}$$
$$K=mgR_E$$
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