Q:

Need for acceleration in particle-antiparticle annihilation?

If an electron and positron are accelerated towards each other, at distances quite far away, there wouldn't be any significant electrostatic attraction, hence they need to be accelerated. But when they do come close, the Coulomb force is significant. So why do we accelerate the particle antiparticle pair, when they get attracted by electromagnetic forces? What is the need for getting them to collide at high speeds? Doesn't annihilation occur when an electron comes in the field of the positron?

A:

Pairs of charged particles and/or objects attract via the $Q_1Q_2/R^2$ Coulomb's law. This is a classical approximation that quantifies how their velocities are changing when the objects are large or distances are much longer than the Compton wavelength etc.
When the particles get really close, there are new effects that are neglected by the laws of classical physics such as Coulomb's law. They must be calculated using the so-called quantum mechanics - or, when the velocities are high or creation/annihilation takes place, quantum field theory is enough.
Quantum field theory implies that the repulsion or attraction isn't the only thing that the electron-positron pair may do. One of the charged particles may also emit a photon and go off-shell for a while, before it emits another photon and goes to the negative, opposite energy-momentum than its partner particles. Then they may annihilate. This is only possible due to the Heisenberg uncertainty – and may borrow an energy for a while. 
That's why the particles may annihilate even though their classical trajectories wouldn't even hit each other. In quantum mechanics, the electron and positron trajectory merge into a single trajectory that is bended backwards in time (positron is an electron going backwards in time) and from which two photons are emitted. Such histories have to be summed over and they contribute to processes that are possible. In QFT, creation and annihilation is unavoidably possible as long as charge conservations are respected.

