Average velocity

Prerequisites

Velocity is the answer to the question: How fast are you changing your position?  It's basically asking for a comparison of where you are at two different times and makes the rate of change quantitative.  To make sense out of this, let's first write it as a words equation:

Average velocity = (How far did you move?) / (How long did it take you?)

That way, doing either a bigger distance in the same time or the same distance in a shorter time both give a larger number to the velocity and agree with our sense of what it means to move faster. (We show the icon of the dog here since we want you to "make sense" of the velocity equation, pulling the picture of the spotted dog out of a picture with lots of spots. Click on the image to see what we mean.)

Warning: Although the word equation helps with making conceptual sense of what's going on with velocity, it doesn't capture everything we are thinking about when we talk about velocity.  We mean velocity to be a vector with "how far did you move" really standing for "what was your vector displacement"?  This allows us to do much more with velocity than the word equation does.

We call this the average velocity because it only pays attention to the beginning and the end — how big the change was in your position — and not how did you get from your starting point to the finishing point.  Thus, in the Garfield cartoon below (keeping the "dog" metaphor), the fact that Garfield kicked Odie back to his starting point means that his average velocity was 0 — despite having moved in the middle, because the two motions (to the right and to the left) cancelled each other out.

We express this in symbols by putting an angle brackets around the velocity to indicate "average" — like this: $\langle v \rangle$.  (In some texts, an average is indicated by putting a bar over the variable, but since we are already putting a vector arrow over a variable to indicate it has direction, this would get too messy.) As discussed in the page Values, change, and rates of change, we will use the symbol Δ to mark when we mean a change in something.  The equation defining average velocity in symbols then becomes (thinking about a velocity in 2D or 3D):

$$\langle\overrightarrow{v}\rangle = \frac{\Delta\overrightarrow{r}}{\Delta t} $$

If we are more explicit about the initial and final positions and times, we might want to write this as

$$\langle\overrightarrow{v}\rangle = \frac{\overrightarrow{r_f} - \overrightarrow{r_i}}{t_f - t_i} $$

where the "i" subscript means "initial" and the "f" subscript means "final"; so for example, ti means the starting (initial) time.

If we multiply both sides of our defining equation by the time interval, we can get a better sense of what the average velocity means:

$$\Delta\overrightarrow{r} = \langle\overrightarrow{v}\rangle \Delta t$$

So if you moved a distance $\Delta\overrightarrow{r}$ in a time $\Delta t$, the average velocity is that constant velocity that you would have to move to go that distance in that time.  Of course you might not have moved with a constant velocity in that time interval.

Dimensionality of velocity

Since velocity is a ratio of a distance (dimensionality L) to a time (dimensionality T), it has dimensionality L/T:

[v] = L/T.

Average velocity graphically

From our analysis of derivatives and integrals, we can see how position-time and velocity-time graphs relate to each other.  Let's work in 1D so it's simpler. In more dimensions we would use similar equations for the y and/or z coordinates. The basic pair of equations are

$$\langle v \rangle = \frac{\Delta x}{\Delta t}  $$

$$\Delta x = \langle v \rangle \Delta t$$

We use the first equation to interpret v on a position (x-t) graph. 

The average velocity over a time interval is the change in position (the rise — shown in blue) divided by the time interval (the run — shown in red). So the velocity is the slope of the hypotenuse of the little triangle (with red-blue-black sides).  If we make the time interval small, the slope becomes the slope of the tangent to the position curve and that's the value we put at that time on the velocity graph—- at the time half-way between $t_1$ and $t_2$.

If we want to go back — from the velocity graph to the position graph, we use the second equation. The average velocity times the time interval is the change in position.  A situation is shown in the graph below at the left where the velocity is plotted as a function of time (solid black line) and is changing. Let's consider what the average velocity might be between the times $t_1$ and $t_2$. By what we learned about the integral, we know that the displacement ($\Delta x$) is the integral — the little bits of $v$ times $\Delta t$ added up -- so it is the area under the curve, shown in the middle graph in blue.

Since the average velocity is a constant over that time interval, we have to adjust the position of the constant v line so that it has the same area under it.  This result is shown at the right. The average velocity line has been slid up and down until the part of the area under the curve that is NOT included (in pink) under the average velocity line is equal to the extra area that IS included (in light blue).  As a result, the area in blue in the rectangle in the last graph determined by the average velocity line (in light and dark blue together) is exactly equal to the area under the middle curve (in dark blue).  These areas (basically a height = velocity times a width = time) are equal to the change in position.

Uniform Motion

If you actually ARE going at a constant velocity, then the average velocity is equal to the (constant) velocity, say v0, and the above equations give a simple expression for the position as a function of time.  There are lots of ways to write this, for example:

$$\langle\overrightarrow{v}\rangle = \frac{\Delta\overrightarrow{r}}{\Delta t} =  \overrightarrow{v_0}$$

$$\Delta\overrightarrow{r} =\overrightarrow {v_0} \Delta t$$

$$\overrightarrow{r_2} - \overrightarrow{r_1} =\overrightarrow {v_0} \Delta t$$

$$\overrightarrow{r_2} = \overrightarrow{r_1} +\overrightarrow {v_0} \Delta t$$

(Notice: Vectors are kind of like dimensions.  You can only add vectors to vectors and you can only equate vectors to vectors.  This means if one side of an equality is a vector, the other side has to be one too.)

You can probably think of lots more ways you could write it — like by opening up the time interval the way we have the change in position. (The last one looks like the stepping rule we discussed when we talked about what a derivative is good for.)

What's the point?

The average velocity equation summarizes an intuitive relationship that just "makes sense".  For example, consider the following problems:

  • If you were to drive north on I-95 for 2 hours (on a Sunday morning when there isn't much traffic) and could average 60 mi/hr, how far would you have gone?
  • Suppose there was traffic and you could only average 30 mi/hr.  How long would it take you to go the same distance?

You probably could answer these without even thinking much about it.  But suppose you were averaging 23 mi/hr.  Now how long would it take you to go the same distance?  Most people can't do this in their head.  The equation summarizes the intuitive relationship you had for the first two questions and allowed you to go to your calculator and helped you make sure that you performed the correct operations on the appropriate numbers to maintain that same intuitive relationship — without the intuition.

Note: Velocity vs. Speed

You may have heard the words "velocity" and "speed" used interchangeably, but in physics, they have different specific meanings.  Velocity is a vector, a quantity with both a magnitude and a direction.  Speed is a scalar, a quantity that is just a magnitude.  So an example of a velocity might be 20 m/s northeast, or 20 m/s in the positive x-direction; an example of a speed might be 20 m/s.

Average velocity is calculated by dividing your displacement (a vector pointing from your initial position to your final position) by the total time; average speed is calculated by dividing the total distance you traveled by the total time.  If you run around a circular 400-meter track in 80 seconds, and end up back at your starting point, your average velocity is zero (as discussed above), but your average speed is 5 m/s.

Joe Redish and Ben Dreyfus 2/2/15

 

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