  Geodesic equations of the vacuum C-metric are derived and solved for various
cases. The solutions describe the motion of timelike or null particles with
conserved energy and angular momentum. Polar, nearly-circular orbits around
weakly accelerated black holes may be regarded as a perturbation of circular
Schwarzschild geodesics. Results indicate that circular Schwarzschild geodesics
of radius $r_0>6m$ are stable under small uniform accelerations along the
orbital plane. These stable orbits undergo small oscillations around $r_0$,
behaving like a harmonic oscillator driven by a periodic force plus another
constant force. Circular orbits with axis parallel to the direction of black
hole acceleration are also considered. In this case an algebraic relation
expressing the condition of stability is obtained. This refines the stability
analysis done in previous literature. We also present an analysis of radial
geodesics along the poles. There exist a solution where a particle remains at
unstable equilibrium at a fixed distance directly behind the accelerating black
hole. Examples of numerical solutions are presented for other more general
cases.
