Velocity as a Rate

Throw a ball straight up. Its height in metres after t seconds is h(t) = 30t - 5t^2. After 1 second it is 25 metres up, and after 2 seconds 40 metres. Over that second its height grew by 15 metres, so its average velocity was 15 metres per second.

Average velocity is the average rate of change of height: change in height over change in time, (h(2) - h(1))/(2 - 1). Metres on top and seconds underneath give metres per second. The sign carries the direction. From t = 4 to t = 5 the height drops from 40 to 25, so the average velocity is -15 metres per second: negative means the ball ended lower.

Velocity is not speed. From t = 2 to t = 4 the height goes from 40 to 40, so the average velocity is 0, but the ball did not stay put: it climbed to 45 metres at t = 3 and fell back. It travelled 10 metres in 2 seconds, an average speed of 5 metres per second. Velocity compares only where the ball starts and ends; speed counts every metre travelled and is never negative.