Average speed and average velocity are not the same quantity, and they are calculated differently. This is not just because velocity is a vector and speed is not; it has to do with distance and displacement. On this page we'll show you how to calculate both average speed and average velocity, and explain why they're different. Average Speed Average speed is not a vector quantity. It has no direction. The simplest way to find average speed is to divide the total distance travelled by the total time taken. Have a look at the path followed by the child shown below. She has travelled a total distance of 12 metres, and has stopped 3 metres from where she began. Let's suppose she covered each leg of her journey in one second. This means that the total time for her trip was Her average speed is the total distance she covered, divided by the total time taken. Since she covered a total distance of Which direction she went, or where she ended up, is unimportant, as long as we know the sum of all the distances she travelled, and how long it took. Average Velocity Average velocity is based on where she ended her trip; specifically, how far she is from her starting point. This distance from the starting point is called displacement, and it's a vector quantity. In the picture above, you can see that the child's displacement is 3 metres right, from where she started. (Notice that displacement comes with a direction). Her average velocity is based on this quantity, not what she did to get there. Since her displacement from where she started was 3 metres to the right, and this trip took a total of 6 seconds, her average velocity was 3 m divided by 6 s, or 0.5 m/s right. Again, notice that the average velocity does not depend on where you went before you got to where you ended up. All that matters is how far you ended up from where you started (displacement), and how long it took. Here's another example to make that clearer. Path 1: The average speed was 30 m divided by 6 seconds, or 5 m/s. The average velocity was 12 metres divided by 6 seconds, or 2 m/s (in a direction that's approximately 110° from vertical). Path 2: The average speed was 18 m divided by 6 seconds, or 3 m/s. The average velocity was 12 metres divided by 6 seconds, or 2 m/s (in a direction that's approximately 110° from vertical). Both trips have the same average velocity despite very different paths, because they ended at the same spot (displacement) and took the same amount of time to get there. Now have a look at the next example, below. The child has taken a short trip, covering a total distance of 15 metres. Again, let's suppose that each leg of the trip took 1 second, making the total time for the trip 7 seconds. Average Speed: The total distance covered was 15 metres. The total time taken was 7 seconds. The average speed was 15 divided by 7, or about 2.1 m/s. Average Velocity: The displacement is 0, since she ended up back where she started. The total time was 7 seconds. The average velocity was 0 divided by 7, or 0 m/s. Despite the trip moving at various speeds, because it ended up at the starting point, the average velocity was zero. This will always be true when the final displacement is zero. Here's a similar situation shown graphically: The position graph shows a trip 'outwards' that covers 6 metres in the first 3 seconds. This is followed by no movement for 3 seconds. The final leg is a trip 'inwards' that covers 6 metres in 3 seconds. On the velocity graph, this is equivalent to a velocity of +2 m/s for 3 seconds, a velocity of 0 m/s for another 3 seconds, and a velocity of -3 m/s in the final 3 seconds. The average speed for this trip was the total distance covered (12 metres) divided by the total time (9 seconds), or about 1.3 m/s. The average velocity for this trip was the displacement (0 m), divided by the total time (9 seconds), or 0 m/s. Graphically, the average velocity will always equal zero when the position graph returns to zero, indicating zero displacement. Have a look at the velocity graph at the right and see if you can recognize when the average velocity will be zero. You may know that you can determine the displacement from a velocity graph by working out the area between the graph and the horizontal axis. In the first segment of the graph, the displacement can be found by working out the area of the rectangle. It is the length (3 seconds) multiplied by the width (+2 m/s), which gives +6 metres. In the second segment of the graph, the displacement can be found by working out the area of another rectangle. It is the length (3 seconds) multiplied by the width (-2 m/s), which gives -6 metres. The displacements add to a total displacement of zero metres. Whenever the segments above and below the horizontal axis on a velocity graph have the same areas, the displacements will add to zero, and the average velocity will be zero. Of course, the regions above and below do not have to be identically shaped as they are here; they merely need to have the same area. Here's a word problem to illustrate the ideas of average speed and average velocity a different way: At a track meet at Fairview High School, a runner does one complete 800 metre lap in 1.5 minutes. Assuming she starts and stops at the same point: a. What was her average speed during this lap? b. What was her average velocity for this lap? First let's change the units of time to seconds. 1.5 minutes is the same as 90 seconds. a. The total distance covered was 800 metres. The total time was 90 seconds. The average speed was 800 divided by 90, or about 8.9 m/s b. The total displacement was zero. She returned to the starting point! The total time was 90 seconds. The average velocity was 0 divided by 90, or 0 m/s |