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| Composition Setup |
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The time rate of change of position. |
Mathematical Definition
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}{composition-setup} {excerpt} The time rate of change of [position|position]. {excerpt} h4. Mathematical Definition {latex}\begin{large}\[ \vec{v} = \frac{d\vec{r}}{dt}\]\end{large}{latex} h4. Representing Velocity Graphically Velocity 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]. h4. Average Velocity h6. 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} |
Representing Velocity Graphically
Velocity 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.
Average Velocity
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Mathematical 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
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\begin{large}\[ f(t)\]\end{large}{latex}
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is:
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}\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 special case of velocity, this expression |
.
In the special case of velocity, this expression becomes:
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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
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formally
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split
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the
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numerator
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into
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three
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integrals
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and
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make
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a
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change
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of
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variables
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in
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each
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of
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the
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integrals.
...
Noting
...
that
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(by
...
the
...
chain
...
rule):
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}\begin{large}\[ dx = \frac{dx}{dt}\:dt\]\end{large}{latex} |
with
...
the
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corresponding
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expressions
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for
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dy
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and
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dz
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,
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we
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have:
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}\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
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integrals
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are
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extremely
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simple,
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and
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lead
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to
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the
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very
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simple
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final
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expression:
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}\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,
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the
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average
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velocity
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is
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simply
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the
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total
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change
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in
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for
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a
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trip
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divided
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by
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the
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total
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elapsed
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time
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for
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the
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trip.
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A
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One-Dimensional
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Example
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Consider
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an
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example
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of
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one-dimensional
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motion.
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Suppose
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a
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student
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rushes
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from
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their
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dorm
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to
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the
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physics
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building
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in
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2
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minutes.
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After
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spending
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4
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minutes
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turning
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in
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their
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homework,
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the
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student
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hurries
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to
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the
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cafeteria
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in
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2
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minutes.
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The
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student
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eats
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lunch
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for
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12
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minutes,
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then
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walks
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to
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the
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library
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in
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6
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minutes.
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During
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which
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portion
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of
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the
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trip
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was
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the
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student
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moving
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the
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fastest?
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For
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simplicity,
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imagine
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a
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school
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where
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all
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these
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buildings
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are
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on
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the
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same
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street.
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The
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street
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runs
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east
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to
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west.
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Suppose
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that
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the
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physics
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building
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is
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two
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blocks
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east
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of
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the
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dorm,
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the
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cafeteria
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is
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one
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block
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west
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of
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the
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dorm,
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and
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the
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library
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is
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three
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blocks
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east
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of
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the
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dorm.
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Before
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performing
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any
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calculations
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to
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characterize
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this
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trip,
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it
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is
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necessary
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to
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set
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up
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a
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coordinate
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system.
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One
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possibility,
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which
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we
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will
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use
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in
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this
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example,
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is
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shown
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below.
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:
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!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
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see
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how
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this
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equation
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works,
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consider
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the
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first
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part
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of
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the
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student's
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trip.
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In
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that
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part,
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the
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student
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moved
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from
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the
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dorm
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to
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the
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physics
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building
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in
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a
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time
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of
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2
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minutes.
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To
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evaluate
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the
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average
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velocity
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for
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this
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part,
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we
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simply
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substitute
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into
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the
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equation:
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}\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
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we
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have
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used
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the
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subscript
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"p"
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to
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stand
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for
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the
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physics
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building
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and
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"d"
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for
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the
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dorm.
| Note | |||
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| } 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: {
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| Warning |
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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):
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{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
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to
...
note
...
here
...
is
...
that
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our
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answer
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has
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come
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out
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with
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a
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negative
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sign.
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For
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the
...
first
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leg
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of
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the
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trip,
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the
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student
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has
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a
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velocity
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of
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+
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1
...
block/min,
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and
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for
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the
...
second
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leg,
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a
...
velocity
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of
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-
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1.5
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blocks/min.
...
These
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signs
...
indicate
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the
...
direction
...
of
...
the
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students
...
motion,
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just
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as
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the
...
sign
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of
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the
...
position
...
difference
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did.
...
When
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reporting
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average
...
velocities,
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it
...
is
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a
...
good
...
practice
...
to
...
explicitly
...
give
...
the
...
meaning
...
of
...
the
...
signs,
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so
...
that
...
people
...
do
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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} |