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Composition Setup
 Excerpt
The time rate of change of position. 
Mathematical Definition
 Latex
 Wiki Markup
{composition-setup}{composition-setup}
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h1. Velocity
{excerpt}
The time rate of change of [position|position]. {excerpt}
\\

h3. {toggle-cloak:id=mathdef} Mathematical Definition

{cloak:id=mathdef}
{latex}\begin{large}\[ \vec{v} = \frac{d\vec{r}}{dt}\]\end{large}{latex}
\\
{cloak}

h3. {toggle-cloak:id=consvel} 1-D Motion with Constant Velocity

{cloak:id=consvel}

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

{cloak:id=adutil}
Velocity 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.
{note}
\\
{cloak:adutil}

h4. {toggle-cloak:id=util1d} Utility of the One-Dimensional Case

{cloak:id=util1d}
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.

\\
{cloak:util1d}

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

{cloak:id=consgraph}
Looking at the equation above, it should be clear that the mathematical definition of constant velocity is equivalent to the formula for 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.

\\
{cloak:consgraph}

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

{cloak:id=ch1}
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:*
{cloak:id=ans1}
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.
{cloak:ans1}\\

\\
{cloak:ch1}

h4. {toggle-cloak:id=consvelrel} Relevant Examples

{cloak:id=consvelrel}
{contentbylabel:constant_velocity,example_problem|operator=AND|showSpace=false|showLabels=true|excerpt=true|maxResults=50}
{cloak:consvelrel}
{cloak:consvel}

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

{cloak:id=avv}

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

{cloak:id=avvdef}
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}
Representing Velocity Graphically
Velocity is commonly represented graphically in several ways:
Average Velocity
...
Mathematical Definition
...
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}
.
\\

In the 
. 
In the special case of velocity, this expression becomes:
 Latex
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
...
 for 
...
 a 
...
 trip 
...
 divided 
...
 by 
...
 the 
...
 total 
...
 elapsed 
...
 time 
...
 for 
...
 the 
...
 trip.
...
A One-Dimensional 
...
Example
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.
Image Added
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


!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}

 where t denotes elapsed time.
 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.
We can compare this to the average velocity for the second trip made (from the physics building to the cafeteria):
 Latex
{latex}\\
where _t_ denotes elapsed time.
{note}
{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.
{warning}
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?

{warning}
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