Binding Energy

The binding energy of a stationary body on a planet's surface is the energy needed to move it from the surface to \infty, equal to the magnitude of its surface potential energy, $E_b=\frac{GMm}{R}$. Since the object starts at rest, no kinetic term is involved, and using $GM=g{R}^2$ simplifies this to $E_b=mgR$. Substituting $m=1$ kg, $g=9.8$ m/s^2 and $R=6.4\times 10^6$ m gives $E_b=1\times 9.8\times 6.4\times 10^6=6.27\times 10^7$ J. The option $1.25\times 10^8$ J wrongly doubles the result, applying the escape-energy factor twice. The option $3.13\times 10^7$ J halves it as if using the orbital relation $GMm/2R$. The option $9.80\times 10^6$ J mistakenly uses only $g$ times a single power of ten. This follows the NCERT definition of binding energy. It is worth distinguishing this from the binding energy of an orbiting satellite, which is only $\frac{GMm}{2R}$, exactly half this value, because the orbiting body already possesses kinetic energy and therefore needs less additional energy to escape. The binding energy of a stationary surface body is thus the largest, and supplying precisely this amount as kinetic energy is what defines the escape velocity. A plausibility check confirms the energy equals $\frac{1}{2}m{v}_e^2$ with $v_e=11.2$ km/s, giving the same $6.27\times 10^7$ J, so the answer is internally consistent.