Expedite descends from Latin expedīre, meaning 'to free the feet' (ex- + pēs). Its perfect antonym impede means 'to shackle the feet.' Both trace to PIE *ped- (foot), yielding pedestrian, pedal, podium, tripod, pedigree, and pioneer. Roman soldiers going expedītus had shed their heavy packs for quick movement.
To speed up the progress of; to accomplish promptly. As an adjective (archaic): free from impediment, unencumbered, ready for action.
From Latin expedīre (to free the feet, to disentangle, to make ready, to set free from an impediment), from ex- (out, away from) + pēs / pedis (foot), from Proto-Indo-European *pṓds / *ped- (foot). The original image is military: a soldier whose feet are stuck in a snare or mud is impeded; to expedite is literally to pull the foot free. The same root in reverse gives impede (to put something in the foot's way
Expedite and impede are mirror-image twins from the same Latin root for 'foot.' To expedite = free the feet from a trap; to impede = shackle them. Caesar used expedītus as military jargon — an expeditus miles was a soldier stripped of his pack, carrying only weapons, ready to march fast and fight light (De Bello Gallico I.24).