2009: Problem 5

Topic: Gravity
Concepts: Circular orbits, Conservation of angular momentum, Elliptical orbits


Solution:

By conservation of angular momentum, each satellite has the same angular momentum at any point along its orbit. Thus, we can shift point A to coincide with the apogee of the ellipse and point C to coincide with the perigee of the ellipse.

Recall the velocity of a circular orbit is

v=GMrv=\sqrt{\frac{GM}{r}}

The angular momentum is then

L=mvr=mGMrL=mvr=m\sqrt{GMr}

Since rA>rCr_A>r_C, we have LA>LCL_A>L_C. At the apogee, the velocity vBv_B of satellite BB is smaller than the velocity vAv_A of satellite AA since the ellipse is contained in the AA circle. Thus, we have LB<LAL_B<L_A. Similarly, we find vB>vCv_B>v_C at the perigee since the ellipse contains the CC circle. Thus, LB>LCL_B>L_C. Finally, we obtain LA>LB>LCL_A>L_B>L_C so the answer is A.

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