...
Integration
...
of
...
the
...
Definitions
...
in Special Cases 
General Time Integrals
The defnitions of the acceleration and velocity can be integrated, provided something is known about the functional form of the acceleration. The general procedure is to first find the velocity as a function of time by integrating the acceleration:
| Latex |
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Special Cases [!copyright and waiver^SectionEdit.png!|Motion -- 1-D General (Cases)] h4. General Time Integrals The defnitions of the acceleration and velocity can be integrated, provided something is known about the functional form of the [acceleration]. The general procedure is to first find the velocity as a function of time by integrating the acceleration: {latex}\begin{large}\[ \int_{v_{A}}^{v(t)} dv = \int_{t_{A}}^{t} a(\eta)\:d\eta\]\end{large}{latex} {note}We are using {*}η{*} as the dummy variable of integration on the right because we are using {*}_t_{*} as the (arbitrary) endpoint of the integration, and so {*}_t_{*} is not available as the integration variable. You can just as easily use any other dummy variable.{note} and then use that velocity in the integral: {latex} |
| Note |
|---|
We are using η as the dummy variable of integration on the right because we are using t as the (arbitrary) endpoint of the integration, and so t is not available as the integration variable. You can just as easily use any other dummy variable. |
and then use that velocity in the integral:
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\begin{large}\[ \int_{x_{A}}^{x_{B}} dx = \int_{t_{A}}^{t_{B}} v(t)\:dt \]\end{large} |
Two basic forms of the acceleration are widely useful and so we illustrate the procedure for these as special cases.
Case 1: Constant Acceleration
If the acceleration is constant then the velocity has the form:
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{latex} Two basic forms of the acceleration are widely useful and so we illustrate the procedure for these as special cases. h4. Case 1: Constant Acceleration If the acceleration is constant then the velocity has the form: {latex}\begin{large}\[ v(t) - v_{A} = a\cdot (t-t_{A}) \]\end{large}{latex} |
which
...
is
...
substituted
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into
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the
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next
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integral
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to
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find:
| Latex |
|---|
}\begin{large}\[ x_{B} - x_{A} = \int_{t_{A}}^{t_{B}} \left(a(t-t_{A})+ v_{A}\right)\:dt = v_{A}(t_{B}-t_{A}) + \frac{1}{2}a(t_{B}-t_{A})^{2}\]\end{large}{latex} |
This
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equation
...
is
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the
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basic
...
...
...
...
for
...
the
...
...
...
...
...
...
...
.
| Tip |
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| } Does this agree with the [ ]of the [ Constant Velocity|1-D Motion ( )] []for the special case {*}_a _= 0 {*}?{tip} h4. Case? |
Case 2:
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Acceleration
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Varying
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Linearly
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with
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Position
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The
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procedure
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illustrated
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above
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will
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work
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for
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any
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acceleration
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which
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given
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as
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an
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explicit
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and
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integrable
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function
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of
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time
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(though
...
of
...
course
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some
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integrable
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functions
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require
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more
...
work
...
to
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integrate
...
than
...
others).
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The
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procedure
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must
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be
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modified,
...
however,
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if
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the
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acceleration
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is
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given
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as
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an
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implicit
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function
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of
...
time.
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The
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simplest
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such
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specification
...
that
...
is
...
commonly
...
encountered
...
in
...
mechanics
...
is:
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}\begin{large}\[ a = -\omega^{2}x(t) \]\end{large}{latex} where {*}ω{*} is a (real number) constant. Because the [acceleration] depends _implicitly_ on time through the unknown time dependence of the [position], we must solve for the position by construcing a second-order |
where ω is a (real number) constant. Because the acceleration depends implicitly on time through the unknown time dependence of the position, we must solve for the position by construcing a second-order differential equation:
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differential equation: {latex}\begin{large}\[ \frac{d^{2}x}{dt^{2}} = -\omega^{2} x \]\end{large}{latex} |
Solving
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a
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differential
...
equation
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often
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requires
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specialized
...
techniques,
...
but
...
in
...
some
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simple
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cases
...
(like
...
this
...
one)
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a
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good
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calculus
...
student
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can
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guess
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the
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answer
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based
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upon
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the
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properties
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of
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common
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functions.
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In
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this
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case,
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the
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fact
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that
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the
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trig
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functions
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sine
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and
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cosine
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are
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proportional
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to
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the
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negative
...
of
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their
...
own
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second
...
derivative
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is
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a
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big
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clue
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that
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the
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answer
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can
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be
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given
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in
...
terms
...
of
...
these
...
functions.
...
We
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will
...
explore
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the
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solutions
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to
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this
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differential
...
equation
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in
...
more
...
detail
...
when
...
we
...
learn
...
the
...
...
...
...
...
.
| Note |
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| } The character of the equation changes dramatically if the negative on the right hand side is removed, and sine or cosine is no longer a solution. The answer is still readily guessed by a student familiar with the basic functions of calculus, however. Can you think of a common function which would solve the case when the right hand side is positive? |
Specializations and the Hierarchy of Models
The fact that we will study the two specializations discussed above in detail means that it makes sense to consider them separate models even though they are clearly special cases of the One-Dimensional Motion (General) model. Thus, if you examine the hierarchy of Models, you will see these two special cases listed below the General motion case as One-Dimensional Motion with Constant Acceleration and Simple Harmonic Motion. Of course, we could generate an infinite number of special cases, such as accelerations that are polynomial functions of time, accelerations that are proportional to velocity, and so on. In the hierarchy, we present only the specializations that are commonly taught and used in introductory mechanics. If you take a more advanced course in mechanics, you may want to expand the hierarchy to incorporate new specializations.
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One-Dimensional Motion (General) -
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