Satellite A remains in a stable circular orbit of 2 RE above the earth’s center. If satellite B were to maintain a stable circular orbit of 4RE above the Earth’s center, what velocity must it maintain with respect to satellite 1?

Question

BlackTom AI · Accepted Answer

Orbital speed for a circular orbit is v = sqrt(GM/r), so the speed depends on the orbital radius r as v ∝ r^(-1/2).

1) Relevant knowledge points and concepts
- Gravitational parameter GM (Earth mass times G) is constant for both satellites.
- Circular orbital speed formula: v = sqrt(GM/r).
- The satellite mass cancels out; only the orbital radius matters.

2) Step-by-step reasoning
- Let R_E denote Earth’s radius. Satellite A orbits at r_A = 2 R_E. Satellite B orbits at r_B = 4 R_E.
- Write the speeds: v_A = sqrt(GM / r_A) and v_B = sqrt(GM / r_B).

3) Calculations with intermediate steps
- Substitute radii: v_A = sqrt(GM / (2 R_E)), v_B = sqrt(GM / (4 R_E)).
- Form the ratio v_B / v_A = sqrt( (GM / (4 R_E)) / (GM / (2 R_E)) ).
- Simplify inside the square root: (GM cancels) = sqrt( (1/4) / (1/2) ) = sqrt( (1/4) * (2/1) ) = sqrt(1/2).
- Therefore v_B = v_A * sqrt(1/2) = v_A / sqrt(2).

4) Why the correct answer is correct
Because orbital speed scales with 1/sqrt(r), doubling the orbital radius from 2 R_E to 4 R_E increases r by a factor of 2 and thus reduces speed by factor 1/sqrt(2). The algebra above directly yields v_B = v_A / sqrt(2).

5) Why the other options are incorrect
- v_B = v_A * sqrt(2) would be true if r_B were smaller by factor 2, not larger; it predicts a faster speed at larger radius, which contradicts v ∝ r^(-1/2).
- v_B = v_A / 2 (or v_B = 2 v_A) are wrong because the dependence is by square root, not linear in radius.
- v_B = v_A is wrong because different radii produce different speeds unless radii are equal.

6) Conclusion
The derived relation v_B = v_A / sqrt(2) matches the provided correct choice (option b).

Satellite A remains in a stable circular orbit of 2 RE above the earth’s center. If satellite B were to maintain a stable circular orbit of 4RE above the Earth’s center, what velocity must it maintain with respect to satellite 1?单项选择题

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