Acceleration is the rate of change of velocity. If an object increases its speed from, say, 10 ft/s to 60 ft/s in five seconds, its acceleration is 10 feet per second per second, or 10 feet per second squared (ft/s2).
How did we arrive at that figure? Divide the change in velocity by the change in time. The change in velocity is 60 - 10 = 50 ft/s. The change in time is 5 - 0 = 5 s. Hence, 50/5 = 10 ft/s2.
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Velocity is the speed of an object in any given direction (constant); acceleration measures the change in speed of an object over time.