= Solution
Write $\mu=GM$ and let $v_c=\sqrt{\mu/r_0}$ be the original circular <orbital speed>. An impulse leaves the <orbital angular momentum> unchanged if it is radial: its <angular impulse> $\mathbf r\times\Delta\mathbf p$ is zero. The tangential <velocity> remains $v_c$, and an added radial component $w$ gives
$$
v_c^2+w^2=\beta v_c^2,\qquad \boxed{\Delta\mathbf v=\pm v_c\sqrt{\beta-1}\,\hat{\mathbf r}.}
$$
Thus <kinetic energy> increases even though <angular momentum> does not. For a <Radial kick from a circular Kepler orbit>, the new specific <orbital energy> and squared specific <angular momentum> are
$$
\varepsilon=\frac{\mu}{2r_0}(\beta-2),\qquad j^2=\mu r_0.
$$
The <orbital eccentricity> is $e=\sqrt{1+2\varepsilon j^2/\mu^2}=\sqrt{\beta-1}$, while the <semi-latus rectum> is $p=j^2/\mu=r_0$. Hence
$$
\boxed{r_{\min}=\frac{r_0}{1+\sqrt{\beta-1}},\qquad r_{\max}=\frac{r_0}{1-\sqrt{\beta-1}}\quad(1<\beta<2).}
$$
For $\beta=2$ the trajectory is a <parabolic Kepler orbit> and has no finite furthest radius; its periapsis is $r_0/2$. For $\beta>2$ it is a <hyperbolic orbit>, with the same formula for $r_{\min}$ but unbounded radius:
$$
\boxed{r_{\max}=\infty\quad\text{for }\beta\geq2.}
$$
One must not use the negative denominator in the bound-orbit expression as a negative physical apocentre. On a departing unbound branch the conic's periapsis may already lie in the past; the future minimum from the kick event is then $r_0$. These are ideal conic orbits; a periapsis inside a real planet would cause impact before the ideal continuation.
\Image[/past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2014-radial-kick-and-launch-error-orbits.png]
{title=<Radial impulse>, orbital turning radii and a launch-direction error}
{description=Original dimensionless orbit calculations. Left: a radial kick changes energy while preserving tangential velocity and angular momentum. Middle: fixed-angular-momentum effective potential with bound, marginal and escaping energies. Right: a fixed-speed direction error gives an ellipse with the intended circular orbit's semimajor axis.}
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