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h1. Velocity

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The time rate of change of [position|position]. {excerpt}
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h3. {toggle-cloak:id=mathdef} Mathematical Definition

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{latex}\begin{large}\[ \vec{v} = \frac{d\vec{r}}{dt}\]\end{large}{latex}
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h3. {toggle-cloak:id=consvel} 1-D Motion with Constant Velocity

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h4. {toggle-cloak:id=adutil} The Advantages and the Utility of Constant Velocity

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Velocity isVelocity is a [vector|vector].  Thus, if an object is moving with constant velocity, it is moving at a constant rate and in a constant direction.  For the special case of constant velocity, we can simplify the mathematical definition of velocity to:
{latex}\begin{large} \[ v = \frac{\Delta x}{\Delta t} \] \end{large}{latex}
where it is important to note that, although we have removed the arrow over the _v_, the velocity is _still_ a vector.  An object moving with constant velocity is _necessarily_ moving in only one dimension.  For the case of one dimensional motion, vectors are not usually written with an arrow.
{note}This special case might seem a bit useless, since it is rare for objects to move purely in a straight line with no change in speed.  However, you can describe certain types of complicated motions with acceptable accuracy by breaking them up into smaller segments during which the velocity is basically constant.
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h4. {toggle-cloak:id=util1d} Utility of the One-Dimensional Case

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As with all [vector|vector] equations, the equations of kinematics are usually approached by separation into components.  In this fashion, the equations become three simultaneous one-dimensional equations.  Thus, the consideration of motion in one dimension with acceleration can be generalized to the three-dimensional case.

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h4. {toggle-cloak:id=consgraph} Representing Constant Velocity Graphically

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Looking at the equation above, it should be clear that the mathematical definition of constant velocity is equivalent to the formula forVelocity is commonly represented graphically in several ways:

* On a [velocity versus time graph].
* As the spacing between points in a [motion diagram].
* As the slope of a [position versus time graph.  To see the utility of this correspondence, consider the following plots:

!posvel.gif! !slowvel.gif! !negvel.gif!

Consider the left plot.  The abscissa is time, and the ordinate is position.  This graph is giving the position of some object (called object A) as a function of time.  Looking at the graph, we can see that for each second of time that elapses, the object changes its position by 2 meters.  Thus the slope of the curve is 2 m/s.  Object A, then, is therefore moving with a velocity of \+2 m/s.

Contrast that with the middle plot.  Object B is only changing its position by 1 meter every second.  Thus, it is moving with a velocity of \+1 m/s.

Finally, look at the plot on the right.  Object C is changing its position by \-2 meters every second, and so it has a velocity of \-2 m/s.  Note that objects A and C have the same speed (magnitude of velocity).  The graphs are different, however, because object C has velocity in the negative direction.

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h4. {toggle-cloak:id=ch1} Check Your Understanding

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Suppose that all three of the plots describe objects moving on the same coordinate system.  The positive direction of the system points east, and the negative points west.  Try to describe the motions of the three objects in words, based upon what you have learned from the graphs.

{toggle-cloak:id=ans1} *Answer:*
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There are many ways to describe the motion, but here are some important features.  At the beginning of the motion, A and C are both at the same location, while B is 5 meters east of A and C.  A begins moving east at 2 m/s, B begins moving east at 1 m/s and C begins moving west at 2 m/s.  By the end of the motion (as plotted) A has caught B (at a position 10 m east of where A and C began).  A and B are now 20 meters east of C.
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h4. {toggle-cloak:id=consvelrel} Relevant Examples

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h3. {toggle-cloak:id=avv}Average Velocity

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h4. {toggle-cloak:id=avvdef} Definition

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In mechanics, the term "average velocity" will almost always be used to denote the time-averaged velocity.  The general defnition of the time average of a function
{latex}\begin{large}\[ f(t)\]\end{large}{latex}
is:
{latex}\begin{large}\[ \langle f(t)\rangle_{t} \equiv \frac{\int_{t_{i}}^{t_{f}} f(t)\:dt}{t_{f} - t_{i}} \]\end{large}{latex}
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In the special case of velocity, this expression becomes:
{latex}\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{t_{i}}^{t_{f}} \vec{v} \:dt}{t_{f}-t_{i}}
=\frac{\int_{t_{i}}^{t_{f}} \left(\frac{dx}{dt}\hat{x} + \frac{dy}{dt}\hat{y} + \frac{dz}{dt}\hat{z}\right) \:dt}{t_{f}-t_{i}}
\]\end{large}{latex}
We can now formally split the numerator into three integrals and make a change of variables in each of the integrals.  Noting that (by the chain rule):
{latex}\begin{large}\[ dx = \frac{dx}{dt}\:dt\]\end{large}{latex}
with the corresponding expressions for _dy_ and _dz_, we have:
{latex}\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{x_{i}}^{x_{f}} \hat{x}\:dx + \int_{y_{i}}^{y_{f}} \hat{y}\:dy
+ \int_{z_{i}}^{z_{f}} \hat{z}\:dz}{t_{f} - t_{i}} \]\end{large}{latex}
These integrals are extremely simple, and lead to the very simple final expression:
{latex}\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\vec{r}_{f} - \vec{r}_{i}}{t_{f}-t_{i}} \equiv \frac{\Delta\vec{r}}{\Delta t} \]\end{large}{latex}
Thus, the average velocity is simply the _total_ change in [position|position] for a trip divided by the total elapsed time for the trip.
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h4. {toggle-cloak:id=1davv}A One-Dimensional Example

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Consider an example of one-dimensional motion. Suppose a student rushes from their dorm to the physics building in 2 minutes. After spending 4 minutes turning in their homework, the student hurries to the cafeteria in 2 minutes. The student eats lunch for 12 minutes, then walks to the library in 6 minutes. During which portion of the trip was the student moving the fastest?

For simplicity, imagine a school where all these buildings are on the same street. The street runs east to west. Suppose that the physics building is two blocks east of the dorm, the cafeteria is one block west of the dorm, and the library is three blocks east of the dorm. Before performing any calculations to characterize this trip, it is necessary to set up a coordinate system.  One possibility, which we will use in this example, is shown below.

!coordinate system^lesson1 variant 2.png!

We are now ready to return to our question: in which portion of the trip was the student moving the fastest? The average velocity for a trip in one dimension will be defined as:
{latex}\begin{large} \[ \langle v\rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}{t_{\rm f} - t_{\rm i}} \] \end{large}{latex}
To see how this equation works, consider the first part of the student's trip. In that part, the student moved from the dorm to the physics building in a time of 2 minutes. To evaluate the average velocity for this part, we simply substitute into the equation:
{latex}\begin{large} \[ \langle v_{\rm pd}\rangle_{t} = \frac{x_{\rm p} - x_{\rm d}}{t_{\rm p} - t_{\rm d}} = \frac{ +2\:{\rm blocks} - 0\:{\rm blocks}}{2\:{\rm minutes}} = + 1\:{\rm blocks/min}\]\end{large}{latex}
where we have used the subscript "p" to stand for the physics building and "d" for the dorm.
{note}One item that is worth noting is that physics problems often do not give actual times. Instead, they give elapsed times. In this situation, for instance, we were not told exactly when the student left the dorm (10:00 AM? 12:00 PM?) or when they arrived at the physics building. We were only told that the _difference_ between the times was 2 minutes. (If the student left the dorm at 11:00 AM, they arrived at 11:02 AM). This information is sufficent to find the average velocity. Because it is so rare to be given initial and final times, the velocity equation is often written:
{latex}\begin{large}\[ \langle v \rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}{t}\]\end{large}{latex}\\
where _t_ denotes elapsed time.
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{warning}It is a common source of confusion that the equations of mechanics often use "final" and "initial" as their subscripts. For the trip described at the beginning of this lesson, it is clear that the (overall) initial position is the dorm (_x_ = 0 m) and the (overall) final position is the library (_x_ = + 3 m), yet we have just used the equation for average velocity with the final position taken to be the physics building. The equations are not required to use the overall final and overall initial positions and times. You are free to break up the motion into as many segments as desired, and apply the equation to the beginning and end of each segment. The only requirement is that the position taken for the "initial" one occurs _earlier_ in the motion than the "final" one.
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We can compare this to the average velocity for the second trip made (from the physics building to the cafeteria):
{latex}\begin{large}\[ \langle v_{\rm cp}\rangle_{t} = \frac{x_{\rm c} - x_{\rm p}}{t_{\rm cp}} = \frac{(-1\:{\rm blocks})- (+ 2\:{\rm blocks})}{2\:{\rm minutes}} = -1.5\:{\rm blocks/min}\] \end{large}{latex}
The first thing to note here is that our answer has come out with a negative sign. For the first leg of the trip, the student has a velocity of + 1 block/min, and for the second leg, a velocity of - 1.5 blocks/min. These signs indicate the _direction_ of the students motion, just as the sign of the position difference did. When reporting average velocities, it is a good practice to explicitly give the meaning of the signs, so that people do not have to be familiar with your specific coordinate system to understand the result. Thus, in this case, a more general way to report the student's movement is to say that for the first leg the average velocity was 1 block/min *east*, and for the second leg the average velocity was 1.5 blocks/min *west*. When the direction is included, the sign is *removed*.
{warning}Using both methods of reporting direction together in one statement results in confusion. What would it mean if we reported the student had an average velocity of - 1.5 blocks/min west?
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