How to Find Average Velocity if You Know the Final Velocity

Velocity is the answer to the question: How fast are y'all changing your position?  Information technology'south basically asking for a comparison of where you are at two different times and makes the rate of alter quantitative.  To make sense out of this, permit's first write information technology as a words equation:

Boilerplate velocity = (How far did y'all move?) / (How long did information technology accept you?)

That mode, doing either a bigger distance in the same time or the same altitude in a shorter time both give a larger number to the velocity and agree with our sense of what it means to move faster. (Nosotros show the icon of the canis familiaris here since we want y'all to "brand sense" of the velocity equation, pulling the picture of the spotted domestic dog out of a flick 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'southward going on with velocity, it doesn't capture everything we are thinking about when we talk nigh velocity.  Nosotros mean velocity to be a vector with "how far did yous move" really standing for "what was your vector deportation"?  This allows us to do much more than with velocity than the word equation does.

Nosotros telephone call this the average velocity because it only pays attention to the beginning and the end — how big the alter was in your position — and not how did y'all become from your starting indicate 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 heart, 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 as well messy.) Every bit discussed in the folio Values, change, and rates of alter, nosotros 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):

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

If we are more explicit nearly the initial and final positions and times, nosotros might want to write this equally

$$\langle\overrightarrow{5}\rangle = \frac{\overrightarrow{r_f} - \overrightarrow{r_i}}{t_f - t_i} $$

where the "i" subscript means "initial" and the "f" subscript ways "final"; then for example, t _i ways the starting (initial) time.

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

$$\Delta\overrightarrow{r} = \langle\overrightarrow{5}\rangle \Delta t$$

So if you moved a distance $\Delta\overrightarrow{r}$ in a fourth dimension $\Delta t$, the average velocity is that constant velocity that you would have to motility to go that altitude in that time.  Of grade you might non 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 fourth dimension (dimensionality T), it has dimensionality Fifty/T:

[v] = 50/T.

Boilerplate velocity graphically

From our analysis of derivatives and integrals, we can meet 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 employ 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 utilise the start equation to interpret five on a position (x-t) graph.

The average velocity over a fourth dimension interval is the change in position (the rise — shown in blue) divided by the fourth dimension interval (the run — shown in red). And then the velocity is the gradient of the hypotenuse of the little triangle (with scarlet-bluish-blackness sides).  If we make the time interval small, the gradient becomes the slope of the tangent to the position curve and that'south the value we put at that time on the velocity graph—- at the time half-manner between $t_1$ and $t_2$.

If nosotros desire to go back — from the velocity graph to the position graph, nosotros use the second equation. The average velocity times the fourth dimension interval is the change in position.  A state of affairs is shown in the graph beneath at the left where the velocity is plotted as a function of time (solid black line) and is changing. Let's consider what the boilerplate velocity might be between the times $t_1$ and $t_2$. By what we learned about the integral, we know that the displacement ($\Delta ten$) is the integral — the little bits of $v$ times $\Delta t$ added up -- and so information technology is the surface area nether the curve, shown in the middle graph in blue.

Since the average velocity is a constant over that time interval, we accept to suit the position of the constant v line and so that it has the same area under information technology.  This result is shown at the right. The boilerplate velocity line has been slid up and downwardly until the office of the area nether the curve that is Not included (in pink) under the average velocity line is equal to the extra expanse that IS included (in light blue).  As a result, the expanse in bluish in the rectangle in the terminal graph adamant past the average velocity line (in light and dark blue together) is exactly equal to the expanse nether the heart curve (in night blue).  These areas (basically a height = velocity times a width = time) are equal to the alter in position.

Compatible Movement

If you really ARE going at a constant velocity, and so the average velocity is equal to the (constant) velocity, say 5 ₀, and the above equations give a simple expression for the position as a role of time.  There are lots of ways to write this, for example:

$$\langle\overrightarrow{5}\rangle = \frac{\Delta\overrightarrow{r}}{\Delta t} =  \overrightarrow{v_0}$$

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

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

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

(Detect: Vectors are kind of like dimensions.  You tin can only add vectors to vectors and you lot tin can simply equate vectors to vectors.  This ways if ane side of an equality is a vector, the other side has to be ane besides.)

You can probably think of lots more ways you could write it — similar past opening up the time interval the manner we have the alter in position. (The concluding one looks like the stepping dominion nosotros discussed when we talked almost what a derivative is good for.)

What'due south the point?

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

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

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

Note: Velocity vs. Speed

You lot 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 yard/s in the positive x-direction; an case of a speed might be 20 m/s.

Boilerplate velocity is calculated by dividing your deportation (a vector pointing from your initial position to your final position) by the total time; average speed is calculated past dividing the total distance you traveled by the full time.  If you run around a round 400-meter rails in 80 seconds, and end up dorsum at your starting bespeak, your boilerplate velocity is zero (every bit discussed above), but your boilerplate speed is v k/s.

Joe Redish and Ben Dreyfus 2/2/15

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Source: https://www.compadre.org/nexusph/course/Average_velocity