Discussion
A satellite is injected at an altitude of 350 km above the Earth's surface, with a velocity of 8.0 km/s parallel to the local horizon. (Earth radius=6378 km, \(\mu_{E}\) (GM=Gravitational constant × Earth mass) = 3.986×1014 m3s-2). The satellite
Answer: B
The satellite forms an elliptic orbit.
At injection the radius is \(r = 6378 + 350 = 6728\) km. Compare the injection speed with the circular and escape speeds at that radius:
\[ \begin{aligned} v_{c} &= \sqrt{\mu_{E}/r} = \sqrt{\frac{3.986 \times 10^{14}}{6.728 \times 10^{6}}} \\ &\approx 7.70\ \text{km/s} \\ v_{esc} &= \sqrt{2}\,v_{c} \approx 10.89\ \text{km/s} \end{aligned} \]
The injection speed of 8.0 km/s, horizontal, lies between the two, so the orbit is an ellipse with the injection point as perigee. Exactly 7.70 km/s would give a circle, 10.89 km/s or more would escape, and a speed below circular at this height would make injection the apogee of a lower orbit, which is how a satellite might fall back.
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