| unmigratedcomposition-wiki-markupsetup |
|---|
| Excerpt |
|---|
...
The |
...
time |
...
rate |
...
of |
...
change |
...
of velocity of an object, or alternately the net force on the object divided by the object's mass. |
Mathematical Representation
| Latex |
|---|
velocity.{excerpt} || PageContents || | {toc:style=none|indent=10px} | h2. Mathematical Definition {latex}\begin{large}\[ \vec{a} = \frac{d\vec{v}}{dt} h2. \qquad \mbox{or} \qquad \vec{a}=\frac{\sum \vec{F}}{m} \]\end{large} |
One-Dimensional
...
Acceleration
...
Utility
...
of
...
the
...
One-Dimensional
...
Case
...
As
...
with
...
all
...
...
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.
Useful Digrammatic Representations
Several diagrammatic representations are commonly used to represent accelerated motion.
Deceleration
In physics, the term acceleration denotes a vector, as does velocity. When the acceleration of an object points in the same direction as its velocity, the object speeds up. When the acceleration of an object points in the direction opposite the object's velocity, the object slows down. In everyday speech, we would call the first case "acceleration" and the second case "deceleration". In physics, both cases represent acceleration, but with a different relationship to the velocity.
Constant Acceleration
...
Integration with Respect to Time
...
If acceleration is constant, the definition of acceleration can be integrated:
| Latex |
|---|
\begin{large}\[ \int_{v_{\rm i}}^{v} dv = \int_{t_{\rm i}}^{t} a\: dt \] \end{large} |
For the special case of constant acceleration, the integral yields:
| Latex |
|---|
\begin{large} \[ v - v_{\rm i} = a(t-t_{\rm i}) \] \end{large} |
which is equivalent to:
| Latex |
|---|
\begin{large} \[ v = v_{\rm i} + a (t-t_{\rm i}) \] \end{large} |
We can now substitute into this equation the definition of velocity,
| Latex |
|---|
\begin{large}\[ v case. h4. Differential {latex}\begin{Large} $a = \frac{dvdx}{dt}$\end{Large}{latex} h4. Graphical Besides explicit acceleration graphs, acceleration can be found from the slope of a velocity vs. time graph or from the curvature (concavity) of a position vs. time graph. h4. Through Motion Diagrams In a motion diagram, the acceleration can be estimated by looking at the spacing of the individual snapshots (assuming that the snapshots are separated by equal time intervals). If the spacing is increasing with time, the acceleration is in the direction of motion. If the spacing is decreasing with time, the acceleration is opposite to the direction of motion. ---- h2. Relevant Models {children:page=Two-Dimensional Motion (General)|depth=all} ---- h2. Relevant Examples {contentbylabel:1d_motion}\]\end{large} |
which gives:
| Latex |
|---|
\begin{large} \[ \frac{dx}{dt} = v_{\rm i} + a t - a t_{\rm i} \] \end{large} |
We can now integrate again:
| Latex |
|---|
\begin{large} \[ \int_{x_{\rm i}}^{x} dx = \int_{t_{\rm i}}^{t} \left( v_{\rm i} - at_{\rm i} + a t\right)\:dt \] \end{large} |
to find:
| Latex |
|---|
\begin{large} \[ x - x_{\rm i} = v_{\rm i} (t-t_{\rm i}) - a t_{\rm i} (t-t_{\rm i}) + \frac{1}{2}a( t^{2} - t_{\rm i}^{2}) \] \end{large} |
We finish up with some algebra:
| Latex |
|---|
\begin{large}\[ x = x_{\rm i} + v_{\rm i} (t-t_{\rm i}) + \frac{1}{2} a (t^{2} - 2 t t_{\rm i} + t_{\rm i}^{2}) \] \end{large} |
which is equivalent to:
| Latex |
|---|
\begin{large} \[ x = x_{\rm i} + v_{\rm i} (t-t_{\rm i}) + \frac{1}{2} a (t - t_{\rm i})^{2} \] \end{large} |
...
Integration with Respect to Position
...
The definition of acceleration can also be integrated with respect to position, if we use a calculus trick that relies on the chain rule. Returning to the definition of acceleration:
| Latex |
|---|
\begin{large}\[ \frac{dv}{dt} = a \] \end{large} |
we would like to find an expression for v as a function of x instead of t. One way to achieve this is to use the chain rule to write:
| Latex |
|---|
\begin{large} \[ \frac{dv}{dx}\frac{dx}{dt} = a \] \end{large} |
We can now elminate t from this expression by using the defnition of velocity to recognize that dx/dt = v. Thus:
| Latex |
|---|
\begin{large} \[ \frac{dv}{dx}v = a \] \end{large} |
which is easily integrated for the case of constant acceleration:
| Latex |
|---|
\begin{large} \[ \int_{v_{\rm i}}^{v} v \:dv = \int_{x_{\rm i}}^{x} a \:dx \] \end{large} |
to give:
| Latex |
|---|
\begin{large}\[ v^{2} = v_{\rm i}^{2} + 2 a (x-x_{\rm i}) \] \end{large} |
...
The Utility of Constant Acceleration
...
Stringing together a series of constant velocity segments is not usually a realistic description of motion, because real objects cannot change their velocity in a discontinuous manner. This drawback does not apply to constant acceleration, however. Objects can have their acceleration changed almost instantaneously. Because of this, it is often reasonable to approximate a complicated motion by separating it into segments of constant acceleration.