Interpreting Graphs
Graphical presentation is a quick and easy way to communicate what is transpiring
in a certain situation. There are certain clues you can use to help you
determine/analysis the information in a graph. Since this is so incredibly
important, tonight’s lesson will only deal with the interpretation of graphs.
THE RELATIONSHIPS AMONG THE KINEMATIC QUANTITIES
From an equation standpoint we have defined displacement, velocity, and
acceleration as follows.
These equations provide profound insights into the graphical relationships among
displacement, velocity, and acceleration.
We know the slope of a line is defined as:
For derivatives you have the
expression:
, where the d’s are the same as D’s, only they mean very tiny weenie
yellow polka dot changes in . . .. Thus, in reality the expression for a derivative is
just another way of writing the old slope formula--only for very small changes in y
and x. Consequently, any derivative represents the slope of a graphed function.
In the case of acceleration the graph has velocity on the y-axis and time on the x-axis.
From this we can conclude that the value of acceleration at any point in time can be
determined from the slope of a velocity-time graph at any point in time.
Also, since velocity is the derivative of position (displacement) with respect to
time, we can conclude that the value of velocity at any point in time can be
determined from the slope of a position-time graph at any point in time.
Consequently, you should be able to look at a position-time graph and state what was
going on with the velocity, and observe a velocity-time graph and state what’s
happening with the acceleration.
|
In the position-time graph to the right objects A and B have
exactly the same velocity. The only difference is that object A
has a head start on object B. Both velocities are constant and
positive, even though B starts in negative territory. This
situation might representing the following situation.
Serena and Dillon walk home at 1 m/s. Since Dillon has
recently become enamored with the Japanese culture, he
makes her walk three steps behind him all the way home.
Not only should you be able to describe what happens, but you
ought to be able to construct a velocity-time graph as well. In
this case, lines A and B will be identical on the velocity-time
graph because the slopes are identical. Remember, velocity by
itself tells you nothing about the exact position (we will see
shortly that is does tell the amount that the position changes).
However, by remember that acceleration is the slope of a
velocity-time graph, we can determine the acceleration at any
point. Here, the acceleration will be zero throughout the
duration of the motion, because the velocity is constant. That is,
its slope is unchanging. Therefore, the acceleration-time graph
would appear as shown in the next illustration.
Obviously, things are going to get a little more complicated. For
example, the change in position doesn’t have to remain
constant. The distance covered could increase or decrease with
time. In such cases, the position-time graph would have a
curved line rather than a straight line.
|
|
|
For the illustration at the left, both curves have positive slopes, but
A’s slope is increasing in value and B’s is decreasing. Thus, in
either case the velocity graph is going to have a diagonal line,
since the velocity is changing. The velocity line for A goes from
zero to higher positive values, because the slope of A’s position
line goes from zero to higher positive values (notice it starts flat
and gets steeper. For B, the opposite is true.
A real life scenario might be:
In this case, Dillon (line A) and Serena (line B) are
standing together when he yells “race ya to the dumpster!”
Dillon immediately starts running pretty fast, but can’t
sustain it, even for an instant slowing down practically to
a stop after 4 seconds. Serena, however, is more
competitive and, although she didn’t start as quickly, soon
passes Dillon and reaches the dumpster well ahead of him.
Once there Serena chows down and exclaims, “this is better
and cheaper than the dinner I had before the homecoming
dance.”
It is difficult to discern the exact characteristics of a velocity line
from a curved position-time graph. In truth, you need to use other
information to make an accurate plot. Generally, a rough guess is
OK and you can assume constant acceleration. The only time where
this would not be the case is for harmonic motion--a subject best
left for a session unto its own.
|
The acceleration graphs ought to be pretty easy sine the slope of a straight line is
constant. Notice that even though the velocity of B is always position, the
acceleration is negative.
GOING THE OTHER WAY?
What if you are given an acceleration graph and you need to find a velocity and a
position graph? As we have seen, two different position graphs can yield the same
acceleration graph. How can you be certain that you can arrive at the right answer?
In truth, you can’t unless the problem gives you the initial position and initial
velocity.
The next question could well be, how do you change an acceleration graph to a
velocity graph. Let’s look at the equation for finding velocity from acceleration.
Whenever you see an integral, it should serve as a big reminder that you need to find
the area under the curve--or, more accurately, the area between the graphed line
and the x-axis. This area tells you the amount by which the quantity you want to
find, changed. In the case of an acceleration-time graph the area under the curve
represents the change in the velocity, not the value of the velocity.
The acceleration graph to the right has a number of different accelerations, which
distinguishes it from the previous graphs we have evaluated. To determine the
velocity graph we will look at the area under the curve for each time interval.
|
Now, to construct a velocity-time graph from this data, you need
to know the initial velocity. If none is given, then assume the
initial velocity is zero. For these graphs you will be plotting
points. Since no initial velocity is given then the point at t = 0,
will have a velocity of zero. From there the change in velocity
tells you how much higher or lower the point should be at the
next time interval. For this example, the velocity values
starting at 1 second and increasing sequentially are: 2, 4, 2, 0,
-2, -1, 0, +1, 0, -1. The graph would appear as shown to the right.
|
|
Just to make sure you did things correctly, you can check you work by seeing if the
slopes of the lines would yield the acceleration-time graph you were given
originally.
Finding the position-time graph from the velocity-time graph uses the same
technique as we used to determine the velocity-time graph. Why? Remember the
equation:
When you integrate to find an answer, you use the area under the curve to determine
how much the value of the position changed. Once again, you will measure the area
under the curve in successive time intervals, only this time you are using the
velocity curve.
|
|
|
Just as before, since no initial position was specified, then you assume that the initial
position is zero. Thereafter, you plot the points and play connect the dot with smooth
curved lines.
SIDE NOTES:
In these examples, we only worked with position, velocity, and acceleration.
However, the concepts of derivatives relating to slope and area under the curve
relating to integration, work for any relationship. Therefore, given an equation and
a graph, you should be able to determine the quantity specified by the equation.
The quantities differ, but the analysis process remains the same.
PROBLEMS:
|
