Total Energy Of A Satellite

A satellite in a circular orbit has kinetic energy $K=\frac{GMm}{2r}$ and potential energy $U=-\frac{GMm}{r}$, so the total mechanical energy is $$\begin{gathered} E=K+ \\ cup=\frac{GMm}{2r}-\frac{GMm}{r}=-\frac{GMm}{2r} \end{gathered}$$. The total energy is negative, which is the signature of a bound orbit. Substituting the values, $E=-\frac{4.0\times 10^{14}\times 1000}{2\times 7.0\times 10^6}=-\frac{4.0\times 10^{17}}{1.4\times 10^7}=-2.86\times 10^{10}$ J. The option $-5.71\times 10^{10}$ J forgets the factor of one-half and uses the full potential energy. The positive option $+2.86\times 10^{10}$ J carries the wrong sign and would describe an unbound trajectory. The option $-1.43\times 10^{10}$ J halves the result a second time in error. This matches the NCERT energy analysis of orbital motion. A powerful consequence of these relations is that to raise a satellite into a higher orbit, where the total energy is less negative, one must actually add energy even though the orbital speed there is smaller; the gain in potential energy outweighs the loss in kinetic energy. This is also why atmospheric drag, by removing energy, paradoxically causes a decaying satellite to speed up as it spirals inward to a lower, more tightly bound orbit. A plausibility check confirms $E=-K$ and $E=U/2$, the standard energy relations, and the negative sign correctly indicates the satellite is gravitationally bound.