Probably from Middle English 'bunsen' (to beat), possibly Low German or Dutch — the rebound sense developed in the 16th century.
To spring back from a surface after hitting it; to move up and down repeatedly.
Of uncertain ultimate origin, first appearing in Middle English as bunsen or bounce (to beat, to thump, to knock heavily). Probably related to Low German bunsen (to beat, to thump heavily) and Dutch bonzen (to thump, to bang, to knock), suggesting a West Germanic source of imitative origin, though no secure PIE root has been identified. Some scholars propose a connection to Old French bondir (to leap, to bounce, to resound), itself of debated Germanic origin. The early English sense was percussive — to strike heavily — and the modern sense of springing
The slang use of 'bounce' to mean ejecting someone from a venue dates to the 1870s in American English — a 'bouncer' was someone who physically 'bounced' troublemakers out the door. The financial sense of a check 'bouncing' (being returned unpaid) dates to 1927, from the image of the check springing back from the bank.
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