Velocity is the change in how far you go over time. Meters per second means “how far you have gone, in meters, each second.”
Acceleration is the change in velocity. Meters per second *per second* means “how much your speed, in meters per second, changes each second.”
So if you start at a velocity of 10 meters per second, and your acceleration is 10 meters per second per second, then after one second you will be going at 20 meters per second. After two seconds you will be going at 30 meters per second.
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