Average Velocity

We all have a common understanding of the terms speed and velocity. Here we want to make these a little more precise and give them more mathematical definitions.

Definition:

We define the average velocity as the total displacement during a given time interval, divided by that time interval:
Average velocity = displacement/total time
\[\rm v_{av} = \frac{\Delta x}{\Delta t} = \frac{x-x_0}{t-t_0} \]
In the previous example, if it took 11.2 seconds to go from x₀ (7.5 m) to x (21.1 m), then the average velocity is given by
\[\rm v_{av} = \frac{\Delta x}{\Delta t} = \frac{21.1 m - 7.5 m}{11.2 s} = 1.22 m/s \]
The average velocity is a vector; it has the same direction as the displacement vector. Note that if you go from 7.5 to 21.2 and then back, your displacement is zero, since x = x₀, and therefore your average velocity is zero no matter how fast you do it!

Definition:

We can also define the average speed as the ratio of the distance traveled during a given time interval and that time interval.
Average speed = distance traveled / total time
At first sight, these two definitions may not sound any different from each other. But remember that distance and displacement do not mean the same thing.

So in the above example, if you took 11.2 seconds to go from 7.5 to 21.2 and the same time to go back from 21.2 to 7.5, the average speed will be 1.22 m/s, even though your average velocity is zero.

To find the distance traveled, add the distances in the individual legs. Do not use the absolute value of the displacement if there is more than one leg.

One more example:

I drive from Lansing to Detroit in 1.5 hours and then back to the same point in Lansing in another 1.3 hours. The distance Lansing-Detroit is 90 miles. In that case, my average speed was:
(90 miles + 90 miles)/(1.5 hours + 1.3 hours) = 64.3 mph
Since there was no net displacement in my drive - I arrived at the same point from which I left - the average velocity for my excursion is 0. That's one reason why they post speed limits and not velocity limits on the freeways...

The SI units for both average velocity and average speed are m/s. The dimensionality of both speed and velocity is [L]/[T]