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Unformatted text preview: 1-dimensional motion with
Traditionally, we call the vertical axis the “ -” axis; the horizontal axis, the “ -” axis. Please
note that there is nothing special about these variables. They are not fixed, and they have no
special meaning. If we are graphing, say, a velocity function ( ) with respect to time , then
we do not bother trying to identify ( ) with or with ; in that case, we just forget about
and . This can be particularly important when representing position with the variable , as
we often do in physics. In that case, graphing ( ) with respect to would give us an on
both the vertical and horizontal axes, which would be extremely confusing. We can even
imagine a scenario wherein we should graph a function of a variable such that would
be on the horizontal axis and ( ) would be on the vertical axis. In particular, in MotionLab,
the variable , not , is always used for the horizontal axis; it represents time. Both and
are plotted on vertical axes as functions of the time . 251 APPENDIX: REVIEW OF GRAPHS There are graphs which are not graphs of functions, e.g. pie graphs. These are not of
relevance to this course, but much of what is contained in this document still applies. Data, Uncertainties, and Fits
). Instead of
When we plot empirical data, it typically comes as a set of ordered pairs (
plotting a curve, we just draw dots or some other kind of marker at each ordered pair.
Empirical data also typically comes with some uncertainty in the independent and
dependent variables of each ordered pair. We need to show these uncertainties on our graph;
this helps us to interpret the region of the plane in which the true value represented by a data
point might lie. To do this, we attach error bars to our data points. Error bars are line
segments passing through a point and representing some confidence interval about it.
After we have plotted data, we often need to try to describe that data with a functional
relationship. We call this process “fitting a function to the data” or, more simply, “fitting the
data.” There are long, involved statistical algorithms for finding the functions that best fit
data, but we won’t go into them here. The basic idea is that we choose a functional form,
vary the parameters to make it look like the experimental data, and then see how it turns out.
If we can find a set of Figure 2: An empirical data set with associated uncertainties and a best-fit line. parameters that make the function lie very close to most of the data, then we probably chose 252 APPENDIX: REVIEW OF GRAPHS the right functional form. If not, then we go back and try again. In this class, we will be
almost exclusively fitting lines because this is easiest kind of fit to perform by eye. Quite
simply, we draw the line through the data points that best models the set of data points in
question. The line is not a “line graph;” we do not just connect the dots (That would almost
never be a line, anyway, but just a series of line segments.). The line does not actually need to
pass through any of the data points. It usually has about half of the points above it and half of
the points below it, but this is not a strict requirement. It should pass through the confidence
intervals around most of the data points, but it does not need to pass through all of them,
particularly if the number of data points is large. Many computer programs capable of
producing graphs have built-in algorithms to find the best possible fits of lines and other
functions to data sets; it is a good idea to learn how to use a high-quality one. Making Graphs Say Something
So we now know what a graph is and how to plot it; great. Our graph still doesn’t say much;
take the graph in Figure 4(a). What does it mean? Something called apparently varies
quadratically with something called , but that is only a mathematical statement, not a
physical one. We still need to attach physical meaning to the mathematical relationship that
the graph communicates. This is where labels come into play.
Graphs should always have labels on both the horizontal and vertical axes. The labels should
be terse but sufficiently descriptive to be unambiguous. Let’s say that is position and is
time in Figure 4. If the problem is one-dimensional, then the label “Position” is probably
sufficient for the vertical axis ( ). If the problem is two-dimensional, then we probably need
another qualifier. Let’s say that the object in question is moving in a plane and that q is the
vertical component of its position; then “Vertical Position” will probably do the trick. There’s
still a problem with our axis labels. Look more closely; where is the object at
knows? We don’t know if the ticks represent seconds, minutes, centuries, femtoseconds, or
even some nonlinear measure of time, like humans born. Even if we did, the vertical axis has
no units, either. We need for the units of each axis to be clearly indicated if our graph is really
to say something. W...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.
- Spring '14