Physics

Physics

Problem

Satellite A moves in a circular orbit of radius 4R4R around a planet of mass MM. Satellite B moves in an elliptical orbit with perigee distance 2R2R and apogee distance 8R8R. When Satellite B is at the point in its orbit that is 4R4R from the planet’s center, how does its kinetic energy KBK_B compare with the kinetic energy KAK_A of Satellite A? a) KB=KAK_B = K_A b) There is insufficient information to compare KBK_B and KAK_A. c) KB>KAK_B > K_A d) KB<KAK_B < K_A
Intuitive explanation

The key is recognizing that both satellites pass through the same orbital radius 4R4R, so you can use energy conservation and orbital mechanics to compare their speeds there. Satellite A has constant orbital energy (circular orbit), while Satellite B's total mechanical energy is fixed by its ellipse's semi-major axis. Calculate each satellite's total energy, then use K=Etotal−UK = E_{\text{total}} - U at radius 4R4R to find which is moving faster.

Intuitive explanation

The key is recognizing that both satellites pass through the same orbital radius 4R4R, so you can use energy conservation and orbital mechanics to compare their speeds there. Satellite A has constant orbital energy (circular orbit), while Satellite B's total mechanical energy is fixed by its ellipse's semi-major axis. Calculate each satellite's total energy, then use K=Etotal−UK = E_{\text{total}} - U at radius 4R4R to find which is moving faster.

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