As a verb, to exhaust means to make someone extremely tired or to use something up completely. As a noun, exhaust is the waste gas that comes out of an engine.
It comes from Latin 'exhaurire', meaning 'to draw out, drain', from 'ex-' (out) and 'haurire' (to draw, scoop). The sense of draining something dry became both using up energy and releasing used gases.
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