Orbital speeds around Earth — identify the correct case: For a satellite to remain in a near-circular low Earth orbit, it must be projected close to the first cosmic (circular) velocity. Which of the following statements correctly corresponds to a circular-orbit condition near Earth?

Difficulty: Medium

Correct Answer: 8.04 km/s at a height of about 285 km (circular orbit)

Explanation:


Introduction / Context:
Satellite motion around Earth depends on achieving the correct orbital speed for a given altitude. The “first cosmic velocity” near Earth’s surface is about 7.9–8.0 km/s, rising slightly with altitude for circular orbits. Distinguishing circular, escape, and geostationary conditions avoids common misconceptions in space mechanics.


Given Data / Assumptions:

  • Standard gravitational parameter of Earth assumed; atmospheric drag neglected.
  • Low Earth orbit (LEO) circular speed ≈ 7.9–8.0 km/s at a few hundred kilometres altitude.
  • Escape speed near Earth ≈ 11.2 km/s (neglecting rotation).
  • Geostationary orbit radius ≈ 42,164 km from Earth’s centre (≈ 35,786 km altitude), circular speed ≈ 3.07 km/s.


Concept / Approach:

For a circular orbit, v = sqrt(μ/R), where R is orbital radius. At LEO heights (≈ 200–300 km), v is around 7.8–7.9 km/s; quoting 8.04 km/s at ~285 km is a reasonable rounded engineering value indicating a circular orbit condition. Escape speed (≈ 11.2 km/s) is not circular; geostationary speed is ~3.07 km/s, not 11.11 km/s.


Step-by-Step Solution:

Identify the LEO circular speed: ≈ 7.9–8.0 km/s → option (a) matches.Check geostationary case: 11.11 km/s is far higher than the true ~3.07 km/s → option (b) is wrong.Escape criterion: 11.26 km/s corresponds to escape, not circular → option (c) is wrong.Therefore, only (a) is correct; hence (d) is false and (e) is also false because 7.0 km/s is sub-orbital at low altitude.


Verification / Alternative check:

Back-of-envelope using v = sqrt(μ/R) with μ ≈ 3.986e14 m^3/s^2 and R ≈ (Earth radius 6,371 km + 285 km) gives v ≈ 7.75–7.8 km/s; small rounding differences are acceptable in MCQs.


Why Other Options Are Wrong:

They mix up escape speed, circular speed, and geostationary values or assert an incorrect constant sea-level circular orbit speed.


Common Pitfalls:

Assuming one fixed “orbital speed” for all altitudes; confusing escape speed with circular speed; ignoring atmospheric drag near the surface.


Final Answer:

8.04 km/s at a height of about 285 km (circular orbit)

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