# Can you have a negative average velocity?

I understand that it's possible to have a negative instantaneous velocity, but I'm wondering if it's possible to have an negative average velocity.

For example, let's say you start heading north at a constant 10 mph for an hour. The second hour, you drive south (directly toward the starting point) at 10 mph (i.e -10 mph north). For the entire 2nd hour, your instantaneous velocity is -10 mph north, but after the trip, the average comes to 0 mph since you've stopped right where you started.

So does this mean that a negative average velocity is impossible to achieve?

rcgldr
Homework Helper
What if you continued heading south for a third hour, what would the average velocity be if north is considered positive and south is considered negative?

Gotcha. Thanks. I guess I was assuming that any displacement from the starting point would be positive. But if north were positive and south negative, what would west be?

More importantly there is no good reason any positive quantity can't be thought of as negative, as long as you keep the system consistent (a.k.a. it's fine to change + to - anywhere as long as you also change - to +)

Gotcha. Thanks. I guess I was assuming that any displacement from the starting point would be positive. But if north were positive and south negative, what would west be?

This relates to speed, not velocity. Displacement and speed are non-vector quantities, whereas position and velocity are vector quantities. It is impossible to have a negative speed just as it's impossible to have a negative length or a negative magnitude.

As to your question, if there's an east/west, then you have to introduce a new axis on your coordinate system and define a positive and a negative on that axis.

Nugatory
Mentor
Gotcha. Thanks. I guess I was assuming that any displacement from the starting point would be positive. But if north were positive and south negative, what would west be?

The quick answer is that west could be either positive or negative, but either way it has nothing to do with north/south; we need one number for north/south and another one for east/west.

As long as you're only allowing movements north and south, you're confining yourself to a single straight line, and one number (for example, positive for northwards and negative for southwards) is good enough to completely specify the velocity. That's another way of saying that a line only has one dimension.

The surface of the earth is two-dimensional, meaning that any velocity can be written as the sum of two velocities, the north-south one that we've already discussed and a second east-west one. Let's choose the positive direction for east-west motion to be eastwards (as good a convention as any, and it's irrelevant which one we pick as long as we all agree to use the same one). Now we'd say that a velocity to the northwest is the sum of a positive north-south velocity and a negative east-west one.

If we were talking about an airplane instead of a car, we'd need to introduce a third dimension, with positive velocity corresponding to climbing and negative velocity to descending.

This would be a good time to google about vectors and vector addition, and about the difference between "speed" and "velocity". You cannot have a negative speed or negative average speed. But it's easy to have a negative average velocity - in the example above, crash the airplane somewhere to the southwest of its starting point.

Nugatory
Mentor
Gotcha. Thanks. I guess I was assuming that any displacement from the starting point would be positive. But if north were positive and south negative, what would west be?

The quick answer is that west could be either positive or negative, but either way it has nothing to do with north/south; we need one number for north/south and another one for east/west.

As long as you're only allowing movements north and south, you're confining yourself to a single straight line, and one number (for example, positive for northwards and negative for southwards) is good enough to completely specify the velocity. That's another way of saying that a line only has one dimension.

The surface of the earth is two-dimensional, meaning that any velocity can be written as the sum of two velocities, the north-south one that we've already discussed and a second east-west one. Let's choose the positive direction for east-west motion to be eastwards (as good a convention as any, and it's irrelevant which one we pick as long as we all agree to use the same one). Now we'd say that a velocity to the northwest is the sum of a positive north-south velocity and a negative east-west one.

If we were talking about an airplane instead of a car, we'd need to introduce a third dimension, with positive velocity corresponding to climbing a negative velocity to descending.

This would be a good time to google about vectors and vector addition, and about the difference between "speed" and "velocity". You cannot have a negative speed or negative average speed. But it's easy to have a negative average velocity - in the example above, start the airplane climbing to the northeast, then reverse course, start descending, and crash it somewhere to the southwest of its starting point.

mfb
Mentor
More importantly there is no good reason any positive quantity can't be thought of as negative, as long as you keep the system consistent (a.k.a. it's fine to change + to - anywhere as long as you also change - to +)
This is true for vectors, but some scalar values cannot be negative (or have a completely different meaning there). Think about the distance between two points, for example, or simply the magnitude of a number. Or temperature - while it can be reasonable to assign negative temperatures to some systems, "colder than 0K" does not exist.

Displacement and speed are non-vector quantities

just to clarify, displacement IS a vector quantity whereas speed is not.

jtbell
Mentor
Gotcha. Thanks. I guess I was assuming that any displacement from the starting point would be positive. But if north were positive and south negative, what would west be?

You can talk about displacement and velocity as positive or negative only when you have one-dimensional motion along a line: east-west or north-south or whatever.

When you have two- or three-dimensional motion, you have to use vectors. The concepts of "positive" and "negative" aren't meaningful for a vector as a whole. You specify a vector either (a) using two or three components, which can each be either positive or negative; or (b) using a magnitude which is always positive, and either one or two angles for the direction.

This is true for vectors, but some scalar values cannot be negative (or have a completely different meaning there). Think about the distance between two points, for example, or simply the magnitude of a number. Or temperature - while it can be reasonable to assign negative temperatures to some systems, "colder than 0K" does not exist.

I was referring to vectors, though what I said could have been misleading so I should have explicitly said that.