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(A) For a circular orbit of radius $r = 2R$,the orbital speed $v$ is given by $v = \sqrt{\frac{GM}{2R}}$.For the elliptical orbit,the semi-major axis

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$\sqrt{3/2} \, v_0$

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(A) For a circular orbit of radius $r = 2R$,the orbital speed $v$ is given by $v = \sqrt{\frac{GM}{2R}}$.For the elliptical orbit,the semi-major axis $a$ is the average of the periapsis and apoapsis distances: $a = \frac{R + 2R}{2} = \frac{3R}{2}$.Using the conservation of energy for the elliptical orbit at the apoapsis (distance $2R$): $\frac{1}{2}mv_0^2 - \frac{GMm}{2R} = -\frac{GMm}{2a} = -\frac{GMm}{3R}$.Rearranging the energy equation: $\frac{1}{2}v_0^2 = \frac{GM}{2R} - \frac{GM}{3R} = \frac{GM}{6R}$.Thus,$\frac{GM}{R} = 3v_0^2$.Substituting this into the expression for the circular orbit speed: $v = \sqrt{\frac{GM}{2R}} = \sqrt{\frac{3v_0^2}{2}} = \sqrt{\frac{3}{2}} v_0$.

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