= Solution
Write $\mu=GM$ and $A=\mu m$. For a <circular orbit> of radius $r$, centripetal balance gives $v^2=\mu/r$. Thus $E=mv^2/2-A/r=-A/(2r)$ and $J^2=m^2r^2v^2=Amr$. Since $e=0$, $a=r_0=r$, these agree with the general expressions.
For an outward coplanar <Hohmann transfer>, burn prograde at the inner <circle> to put its radius at the ellipse's <pericentre>, then burn prograde at <apocentre> to circularize at the station's radius. The station must be phased to meet the spacecraft there. With $r_p=R$, $r_a=\alpha R$,
$$
\boxed{a_T=\frac{1+\alpha}{2}R,\qquad e_T=\frac{\alpha-1}{\alpha+1},\qquad r_0=a_T(1-e_T^2)=\frac{2\alpha R}{1+\alpha}.}
$$
<Energy> at the first burn, with $v_0^2=\mu/R$, gives
$$
\frac{v_1^2}{2}-\frac\mu R=-\frac\mu{(1+\alpha)R},\qquad
\boxed{v_1=v_0\sqrt{\frac{2\alpha}{1+\alpha}}.}
$$
<Angular momentum> is conserved along the transfer ellipse, so $v_{2-}=v_1/\alpha$, while the outer circular speed is $v_{2+}=v_0/\sqrt\alpha$. For $\alpha=3$,
$$
\boxed{v_{2-}=\frac{v_0}{\sqrt6},\qquad v_{2+}=\frac{v_0}{\sqrt3}.}
$$
The correct second <impulse> adds $\Delta v=v_{2+}-v_{2-}$. The <reversed circularization burn in a Hohmann transfer>, using the \i[same] <impulse> magnitude, instead leaves
$$
v_w=2v_{2-}-v_{2+}=v_0\left(\frac2{\sqrt6}-\frac1{\sqrt3}\right)>0.
$$
Its magnitude is below the circular speed at $3R$, so that point is the new <apocentre>. Define $q=r_a v_w^2/\mu=(\sqrt2-1)^2=3-2\sqrt2$. For an ellipse at <apocentre>, $q=1-e_w$, giving
$$
\boxed{e_w=2\sqrt2-2\simeq0.8284,\qquad
r_{p,w}=3R\frac{1-e_w}{1+e_w}=3R\frac{3-2\sqrt2}{2\sqrt2-1}\simeq0.2815R.}
$$
The lunar radius is $R/3$. Since $r_{p,w}<R/3$, \b[the osculating <orbit> intersects the <Moon>: the spacecraft crashes before reaching its formal <pericentre>.] The calculation uses impulsive tangential burns and a spherical central gravitational field; it does not assume that a trajectory continues through the solid body.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2011/paper-2-transfer-orbits.png]
{title=Correct and reversed transfer burn}
The original plot uses $R$ as the length unit and $R/3$ as the lunar radius. The initial station position shown is phased for rendezvous: its initial angular lead is $\pi-\pi[(1+\alpha)/(2\alpha)]^{3/2}$, modulo $2\pi$.
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