The minimum energy required to launch a satellite of mass $m$ from the surface of the earth of mass $M$ and ra

Question

The minimum energy required to launch a satellite of mass $m$ from the surface of the earth of mass $M$ and radius $R$ in a circular orbit at an altitude of $2R$ from the surface of the earth is:

Accepted Answer

1. **Initial Energy ($E_i$):** At the surface of the Earth, the satellite is at rest (ignoring Earth's rotation) at a distance $R$ from the center. Its energy is purely potential.
   $$E_i = U_i = -\frac{GMm}{R}$$
2. **Final Energy ($E_f$):** The satellite is in a circular orbit at an altitude $h = 2R$. The orbital radius is $r = R + h = R + 2R = 3R$. The total energy of an orbiting satellite is given by $E = -\frac{GMm}{2r}$.
   $$E_f = -\frac{GMm}{2(3R)} = -\frac{GMm}{6R}$$
3. **Energy Required ($\Delta E$):** The energy required to launch the satellite is the difference between the final total energy and the initial energy.
   $$\Delta E = E_f - E_i$$
   $$\Delta E = \left( -\frac{GMm}{6R} \right) - \left( -\frac{GMm}{R} \right)$$
   $$\Delta E = -\frac{GMm}{6R} + \frac{6GMm}{6R}$$
   $$\Delta E = \frac{5GmM}{6R}$$

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