Kinetic energy deposited into a nanodroplet, cluster, or molecule in a sticking collision with background gas

In processes when particles such as nanodroplets, clusters, or molecules move through a dilute background gas and undergo capture collisions, it is often important to know how much translational kinetic energy is deposited into the particles by these pick-up events. For sticking collisions with a Maxwell-Boltzmann gas, an exact expression is derived, which is valid for arbitrary relative magnitudes of the particle and thermal gas speeds.