A satellite orbits Earth in a circular trajectory at an orbital radius $r_1 = 7000\text{ km}$ with an orbital period of $T_1 = 98.4\text{ minutes}$. What is the orbital period $T_2$ (in minutes) of a second satellite moving in a circular orbit of radius $r_2 = 28000\text{ km}$ around Earth?

Answer Key:

787.2

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Keplerian Law:** $T^2 \propto r^3$. 2. **Stage 2: Radius Ratio:** $r_2 / r_1 = 28000 / 7000 = 4$. 3. **Stage 3: Power Law Scaling:** $(4)^{3/2} = (\sqrt{4})^3 = 2^3 = 8$. 4. **Stage 4: New Period:** $T_2 = 8 \times 98.4 = 787.2\text{ minutes}$. 5. **Stage 5: Final Answer:** 787.2 (acceptable range 785 to 790).

← Prev Question Next Question →