# Negative or not?

1. Dec 6, 2004

### Tabe

I know that you can have a negative acceleration, which would be considered deceleration, but is it possible to have a negative speed? It may sound like a stupid question to some people, but I am completely baffled.

2. Dec 6, 2004

### futb0l

Don't worry :) I have that same misunderstanding when I started.

Velocity and Acceleration is a vector. That means it is a quantity WITH a direction. For example, I can assign the up direction as + and the down direction as -. Then when I drop an apple it will the acceleration = -g (negative), because it is going down. It is NOT possible to have a negative speed, but it is possible to have a negative VELOCITY.

Speed and Displacement is a vector - it is the quantity without direction.

Still don't understand? Try http://www.physicsclassroom.com/Class/1DKin/U1L1b.html

3. Dec 6, 2004

### cepheid

Staff Emeritus
No we cannot have a negative speed...speed is not a vector quantity. It has only magnitude, not direction. Its value indicates the rate at which an object's distance travelled changes with time, but not in what direction.

Velocity, on the other hand, is a vector quantity. To describe an object's velocity, it is not sufficient to indicate only how fast it is going, but also in what direction.

For the special case of motion in a straight line, we can define a coordinate system such that the line along which the object is travelling is one of the coordinate axes that we have defined (say, the x-axis for example). Then, full blown vectors are not strictly required to describe the velocity...an algebraic scalar (a number with a sign) would be sufficient. If an object is travelling at 10 m/s on the x-axis, then what we call the scalar component of the velocity in the x-direction $v_x$ is:

$$v_x$$ = 10 m/s if it is travelling in the positive x-direction

$$v_x$$ = -10 m/s if it is travelling in the negative x-direction

Since the object is travelling in a straight line, it obviously does not have components in any other direction (other than x!). So we can forget about the x and describe the object's velocity as v, a scalar that has an absolute value equal to the magnitude of the velocity vector, but unlike the magnitude, also has a sign that indicates the directional "sense" (+ or -) of the object's motion along the x-axis. We can write the velocity as

$$v$$ = 10 m/s if it is travelling in the positive x-direction

$$v$$ = -10 m/s if it is travelling in the negative x-direction

If you like, you can use the full blown vector notation instead:

$$\vec{v} = \text{(10 m/s)}\hat{i}$$

(travelling in postive x-direction)

OR

$$\vec{v} = -\text{(10 m/s)}\hat{i}$$

(travelling in negative x-direction).

One point of confusion to watch for (that only arises when the motion is confined along one axis and we choose to drop the x subscript). When using the scalar component notation, v indicates both magnitude (10m/s) and direction, which means it can be either postive or negative. In contrast, when using the vector notation, the same symbol "v" is actually the magnitude of the velocity vector:
$$v = |\vec{v}|$$
so it is always positive (magnitudes are always postive). The direction is instead given by the sign on the unit vector:
$$\pm \hat{i}$$.

4. Dec 6, 2004

### Tabe

Ok, thanks, that explains a lot. It also answers some of the other questions that I had about velocity.