Graphical Analysis
of Motion
By: Mike Maloney
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Graphing Motion
• A common way in physics to describe
different phenomena is with graphs.
Scientists often take data, plot the data,
and they analyze the shape of the
graph to try to determine relationships
between things.
• Our first goal this year will be to
identify, describe, and analyze motion
using graphical techniques.
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Mike Maloney © 2004
Position vs. Time (s vs. t)
• One way to graphically describe the
motion of an object is to plot a graph of
its Position vs. Time.
• It is common practice to put Time on
the horizontal-axis of a graph.
• So we will put Position on the
vertical-axis.
• A position vs. time graph may look
something like this …
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Mike Maloney © 2004
s vs. t
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Mike Maloney © 2004
s vs. t
• From this graph we can tell a few things
•
•
•
•
about the motion of our object.
First we can tell the position of the object at
any point in time by reading it off the graph.
If the graph increases, the object moves in
the (+) direction.
If the graph decreases, the object moves in
the (-) direction.
What does it mean if the graph goes into the
negatives?
– The position is negative, or you might say you are
behind what you called the “0” position.
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Mike Maloney © 2004
Interpreting an “s vs. t”
Now let’s can see what “s vs t” plots tell us about
motion.
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Mike Maloney © 2004
Interpreting an “s vs. t”
We know that we can read the position of an object
directly from a “s vs t” plot.
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Interpreting an “s vs. t”
• At 1.5 seconds, where is the object?
• Between 8 and 9 meters.
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Mike Maloney © 2004
Interpreting an “s vs. t”
We can also determine the change in position by
reading values from the graph.
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Mike Maloney © 2004
Interpreting an “s vs. t”
How far does it move from 1 to 3 seconds?
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Mike Maloney © 2004
Interpreting an “s vs. t”
9
• How far does it move from 1 to 3 seconds?
• s = 14 – 5 = 9 meters.
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Mike Maloney © 2004
Slope of “s vs. t”
Now let’s try do determine some not so easy to see
data from our “s vs t” plot.
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Mike Maloney © 2004
Slope of “s vs. t”
• How do you find the slope of a line?
• change in y / change in x
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Mike Maloney © 2004
“s vs. t”
• So what would the slope of a “s vs t” plot give us?
• change in y / change in x
• change in s / change in t or s / t
• the rate of change of position, or the velocity.
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Mike Maloney © 2004
So far … (s vs. t)
• A position vs. time plot tells us the position (s) of an
object at any point in time
• It can either be positive or negative
• Positive (+): positive of where you started
• Negative (-): negative of where you started
• The change in “y” on a position vs. time plot from one point
to another gives us the displacement (s)
• Going Up (+change): moving in the positive direction
• Going Down (-change): moving in the negative direction
• The slope of the position vs. time plot tells us the velocity
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Mike Maloney © 2004
Velocity vs. Time (v vs. t)
• Another way to graphically describe
motion is to make a plot of the velocity
of an object versus time.
• Again we will put Time on the
horizontal-axis, and Velocity on the
vertical-axis
• A plot of a velocity vs. time graph may
look like this …
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Mike Maloney © 2004
v vs. t
Velocity vs. Time
10 m/s
8 m/s
6 m/s
4 m/s
2 m/s
0 m/s
-2 m/s
-4 m/s
-6 m/s
-8 m/s
-10 m/s
0s
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1s
2s
3s
4s
Mike Maloney © 2004
5s
6s
7s
v vs. t
From this graph we can tell the velocity of an object
at any point in time.
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Mike Maloney © 2004
v vs. t
Notice that this graph has points both positive and
negative.
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Mike Maloney © 2004
v vs. t
• What does the positive part of the graph indicate?
• The times when the velocity was positive, or when the
object was moving in the positive direction.
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Mike Maloney © 2004
v vs. t
• What about the negative part of the plot?
• The times when the velocity was negative, or the
object was moving in the negative direction.
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v vs. t
As the plot increases in the positive direction, the
speed of the object increases.
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Mike Maloney © 2004
v vs. t
As it goes down in the positive part of the graph, the
speed of the object decreases.
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Mike Maloney © 2004
v vs. t
As the plot goes down in the negative section, the
speed also increases, but in a negative direction.
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Mike Maloney © 2004
v vs. t
This would be symbolic of increasing a car’s speed
in reverse.
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Mike Maloney © 2004
v vs. t
As the plot goes up in the negative section, the
speed decreases in the negative direction.
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Mike Maloney © 2004
v vs. t
This would be symbolic of slowing down in reverse.
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Mike Maloney © 2004
Speed and Velocity
• This brings up a common confusion in
physics … why do we have speed and
velocity?
• Speed is just the number part, or the
size of the velocity, it is what the
speedometer in you car reads.
• Velocity not only tells you how fast, but
also in what direction you are going.
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Mike Maloney © 2004
So far … (v vs. t)
• A velocity vs. time plot tells us the velocity of an
object at any point in time
• It can either be positive or negative
• Positive: (+) velocity [+direction]
• Negative: (-) velocity [-direction]
•
•
•
•
Going Up (+part): increasing positive speed, velocity goes up
Going Down (+part): decreasing positive speed, velocity goes down
Going Up (-part): decreasing negative speed, velocity goes up
Going Down (-part): increasing negative speed, velocity goes down
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Mike Maloney © 2004
v vs. t
• So we can read the velocity of an object
directly off the chart.
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Mike Maloney © 2004
v vs. t
• Like in other graphs, we can also talk
about its slope.
• What is the slope of a “v vs. t” graph?
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Mike Maloney © 2004
v vs. t
• Slope = change in y / change in x
• = change in velocity / change in time
• Slope = v / t = acceleration
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Mike Maloney © 2004
6
v vs. t
velocity (m/s)
5
4
3
2
1
0
0
1
2
3
4
5
6
time (s)
• The area underneath the curve gives us
something also.
• How would we find the area underneath the
curve from 1 sec to 3 sec?
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Mike Maloney © 2004
7
6
v vs. t
velocity (m/s)
5
4
3
2
1
0
0
1
2
3
4
5
time (s)
• Area = height x base
= velocity x change in time
• Area = v * t
• that is the equation for displacement (s).
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Mike Maloney © 2004
6
7
a vs. t
• We can extrapolate our ideas on position
and velocity graphs to acceleration vs. time
plots.
• The Area under the curve in an a-t graph is
– (Acceleration) · (time)
– a · t
– Which is change in velocity (v).
• The slope (a/t) actually gives us something
called the jerk, but we don’t have to worry
about that in this class.
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Mike Maloney © 2004
In Conclusion
• Slope of “s vs. t” plot is velocity
• Slope of “v vs. t” plot is acceleration
• Area under “v vs. t” plot is displacement
• Area under “a vs. t” plot is  velocity
• Now lets try some …
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Mike Maloney © 2004