Kevin S. Huang

2010: Problem 25

Topic: Gravity
Concepts: Circular orbits, Elliptical orbits

Solution:

Recall from the elliptical orbit equations that

v_{\text{apogee}}=\frac{a-c}{b}\sqrt{\frac{GM}{a}}

In our case, we have

R=a-c

2R=a+c

Adding the equations,

3R=2a

a=\frac{3R}{2}

Multiplying the equations,

2R^2=a^2-c^2=b^2

b=\sqrt{2}R

Substituting into the expression for $v_{\text{apogee}}$ ,

v_0=\frac{R}{\sqrt{2}R}\sqrt{\frac{GM}{3R/2}}=\sqrt{\frac{GM}{3R}}

The velocity of a circular orbit of radius $2R$ is

v_1=\sqrt{\frac{GM}{2R}}=\sqrt{\frac{3}{2}}v_0

so the answer is A.