The other response discussing conservation of energy and momentum is basically correct. However, the third particle doesn't need to be an atom! Emitting a photo

The other response discussing conservation of energy and momentum is basically correct. However, the third particle doesn't need to be an atom! Emitting a photon with the right energy and spin will also suffice. Typically the transition rates via photon are extremely low, but they can be intentionally triggered with other photons. This yields the incredibly useful tool called photo-association to create molecules from cold atoms.