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

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Solution:

Conserving angular momentum between the apogee and perigee of the ellipse, we have

Since $2R \leq r_1 \leq 3R$ and $R/3 \leq r_2 \leq R/2$, we can bound

Recall the velocity of a circular orbit of radius $r$ is given by

We can relate $v_b$ to $v_c=\sqrt{GM/R}$ by considering a circular orbit of radius $r_1$. Then

since the ellipse is contained in the circle and $r_1 \geq 2R$. We can also relate $v_a$ to $v_c$ by considering a circular orbit of radius $r_2$. Then

since the circle is contained in the ellipse and $r_2 \leq R/2$. Combining these two equations,

We now go through the possible choices:

• A) $v_b>v_c>2v_a$
  Not correct because $v_b<v_c$ from equation 2.
• B) $2v_c>v_b>v_a$
  Not correct because $v_b<v_a$ from equation 2.
• C) $10v_b>v_a>v_c$
  Correct because $v_a \leq 9v_b < 10v_b$ from equation 1 and $v_c<v_a$ from equation 2.
• D) $v_c>v_a>4v_b$
  Not correct because $v_c<v_a$ from equation 2.
• E) $2v_a>\sqrt{2}v_b>v_c$
  Not correct because $\sqrt{2}v_b<v_c$ from equation 2.

Thus, the answer is C.