Is falling damage affected by falling speed?

Supposed you try launching a party member across a 500 feet wide and 50 feet deep gap with a trebuchet. however, something goes horribly wrong and he’s launched straight down into the pit (let’s just say he was stuck in the bag until it reached the end). this would launch him at about 3X the speed of a normal fall (150 feet per second instead of 55 feet per second). Does he still take only 5d10 falling damage (1d10 per 10 feet), or does he take more like 15d10 damage because he lands at triple speed?