Moving forward rapidly and forcefully, often without stopping, like something rolling quickly in a barrel shape.
From 'barrel,' which comes from Old French 'baril' of uncertain origin. The verb form 'barrel' developed in American English to mean rushing forward, and 'barreling' is the continuous form.
This American verb really took off in the 20th century—you'd never find 'barreling' in Shakespeare! It shows how English creates new verbs by turning nouns into action words, especially when the physical meaning suits the action.
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