A satellite A has twice the mass of satellite B. The satellites orbit a planet with the same orbital radius. The orbital velocity of the satellite A compared to B is:

Question

BlackTom AI · Accepted Answer

Orbital speed for a circular orbit does not depend on the satellite’s mass: v = sqrt(G Mplanet / r).

Relevant concepts:
- Gravitational force provides the centripetal force for circular motion.
- For a satellite of mass m at radius r around a planet of mass M: G M m / r^2 = m v^2 / r.
- The satellite mass m cancels, so v depends only on the planet mass M and orbital radius r.

Step-by-step derivation and calculation:
1) Start from Newton’s law of gravitation and centripetal force balance:
   G M m / r^2 = m v^2 / r.
2) Cancel m from both sides (m ≠ 0):
   G M / r^2 = v^2 / r.
3) Multiply both sides by r:
   G M / r = v^2.
4) Take square root:
   v = sqrt(G M / r).
This expression shows v is independent of the satellite mass. For satellite A (mA = 2 mB) and satellite B (mB) at the same r, both have the same v:
   vA = sqrt(G M / r) = vB.

Why this answer is correct:
Because mass m of the orbiting body cancels in the centripetal balance, doubling the satellite’s mass does not change the required orbital speed at a given radius.

Why the other options are incorrect:
- a (vA = 2 vB), b (vA = 1/4 vB), d (vA = 1/2 vB), e (vA = 1/√2 vB) all imply the orbital speed depends on the satellite mass; that contradicts the derivation above. None follow from v = sqrt(GM/r).

Conclusion:
Therefore vA = vB; the correct choice is c, which matches the provided answer.

A satellite A has twice the mass of satellite B. The satellites orbit a planet with the same orbital radius. The orbital velocity of the satellite A compared to B is:单项选择题

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