= Solution
A <central potential> conserves <angular momentum> and confines a nonradial orbit to a plane. With <reduced mass> $\mu$, the radial motion obeys
$$
E=\frac12\mu\dot r^2+U_{\rm eff}(r),\qquad
U_{\rm eff}=U(r)+\frac{L^2}{2\mu r^2}.
$$
Turning points satisfy $E=U_{\rm eff}$. A radially bounded allowed interval gives bound motion, whereas an interval reaching infinity permits escape. The <energy> sign alone classifies these only with suitable assumptions about $U(\infty)$ and barriers. Circular orbits satisfy $U_{\rm eff}'=0$ and are radially stable when $U_{\rm eff}''>0$. Bound orbits in a general <central potential> need not close: successive periapses can precess.
For attractive inverse-square gravity, $U=-\kappa/r$ and
$$
r=\frac{p}{1+e\cos\phi},\quad p=\frac{L^2}{\mu\kappa},\quad
e^2=1+\frac{2EL^2}{\mu\kappa^2}.
$$
Ellipses have $E<0$, $e<1$, and the attracting centre at a focus; parabolic and hyperbolic escape have $e=1$ and $e>1$. Conservation of <angular momentum> gives equal swept areas in equal times. The inverse-square law additionally yields $T^2=4\pi^2a^3/[G(m_1+m_2)]$. These are Kepler's laws; a harmonic <central potential> is another example with closed bound orbits, while generic central laws produce <precession>.
In a <gravitational assist>, a spacecraft's asymptotic <speed> relative to a planet is approximately unchanged but its direction turns. Transforming to the frame in which the planet has <velocity> $\mathbf V$ gives
$$
\Delta\varepsilon=\mathbf V\cdot(\mathbf u_{\rm out}-\mathbf u_{\rm in}).
$$
The spacecraft can gain or lose orbital <energy>; the planet supplies the compensating <energy> and <momentum>, with a very small <velocity> change because of its large <mass>.
Repulsive Coulomb scattering instead has $U=\kappa_C/r$. For incident relative <kinetic energy> $E$ and <impact parameter> $b$, the hyperbolic orbit gives
$$
b=\frac{\kappa_C}{2E}\cot\frac\theta2,\qquad
\boxed{\frac{d\sigma}{d\Omega}=\left(\frac{\kappa_C}{4E}\right)^2
\csc^4\frac\theta2.}
$$
This <Rutherford scattering> formula follows by equating the incident annulus $2\pi b\,db$ to the scattered solid-angle ring. It assumes classical elastic scattering by a point Coulomb centre; screening, finite nuclear size or quantum effects can change the applicable regime.
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