# Q & A: magnitudes of speed and velocity

Q:
In physics, my teacher asked us if the magnitude of speed and the magnitude of velocity were always the same. I thought they were. She showed how if an object travels from one point, a, to another, b, over a time, t, then the magnitude of speed is (b-a)/t or the change in distance over the change in time. Likewise, she did the same for velocity. (x,i - x,f) / (t,i - t,f). In this case, the magnitudes are the same. Then she asked us what the magnitude of speed and velocity were if the object travels from point a, to b, and then back to a again. She said because the starting point and the end point are the same, both x,i and x,f are 0 and then the average magnitude of the velocity is 0. Here is where I have a hard time accepting what she says. I understand that a velocity is actually made up of two components, a magnitude and a direction vector ( mag * direction ). So, in terms of a one dimensional space, a velocity can only have a direction vector with one component, x (being either +1, -1, or 0). So, when something has a velocity of 0, its describing a vector that has a directional x component of 0. This DOES NOT mean that the magnitude is 0; because anything * 0(x direction vector) is still 0. If anything, the magnitude is indeterminate without more data. She gave us more data, however, she said that it traveled with a magnitude of 10 from point a to b, and with a magnitude of 10 from b back to a, taking a total of 5 seconds. So, wouldn't the average velocities magnitude be equal to the average of the magnitudes of each of the resulting vectors( velocity of a to b, and b to a) components? Therefor, the magnitude of the average velocity would be (10+10)/5 or 4? then that magnitude is multiplied by a 0 direction vector to get 0 velocity?
- Misha (age 22)
Freehold, NJ, 07728
A:
It's all really much simpler than that, but your teacher and you need to be careful about using terms consistently. You both obviously know what's really going on, so you're just getting tangled up with words, especially the teacher.

On a round trip, the average velocity (total vector displacement/ elapsed time) is indeed zero by definition. Therefore the magnitude of the average velocity is zero. That is not the same statement as "the average magnitude of the velocity is 0". The magnitude of the velocity is by definition identical to the speed, which is a scalar quantity, not a vector, and never negative. So in any case like this the average magnitude of velocity or average speed is some positive value.

Again, the key point is that the average of the magnitude is not the same thing as the magnitude of the average.

Mike W.

(published on 09/13/2012)