Graphical Analysis of Motion
There are two main types of graph we will analyze in our study of one-dimensional motion, Position vs. Time graphs and
Velocity vs. Time graphs.
Position vs. Time Graphs
On a Position (y axis) vs. Time (x axis) graph :
•
A straight section represents constant velocity
•
A straight horizontal section represents zero velocity
•
The slope of a straight section = change in position/ change in time = the velocity during that section
•
The slope of the line connecting any two points gives the average velocity that occurred between those two
points in time.
•
A curved section represents a changing velocity (acceleration)
•
To get an idea about the velocity in a curved section, draw (or imagine) a tangent to the curve at the point in
question. The slope of the tangent line is equal to the instantaneous velocity.
•
The way the slope is changing determines the sign of the acceleration. (To check for acceleration, imagine the
slope of a tangent line just before and just after the point in question. If the slope is becoming more positive,
the acceleration is positive. If it is becoming more negative, the acceleration is negative)
•
In summary: Initial and final positions and times can be read directly from graph. Displacements can
be calculated (final position – initial position.) Distance can be calculated (distance is the magnitude of
the displacement for any single section of the graph). Velocity is equal to slope of straight section.
Acceleration can not be easily calculated, but its sign can be determined by the way the slope is
changing.
Velocity vs. Time Graphs
On a Velocity (y axis) vs. Time (x axis) graph :
•
A straight section represents constant acceleration
•
A horizontal section represents zero acceleration (constant velocity)
•
The slope of a straight section = change in velocity/ change in time = the acceleration during that section
•
A curved section represents a changing acceleration (changing slope means changing acceleration)
•
To get an idea about the acceleration in a curved section, draw a tangent to the curve at the point in question.
The slope of the tangent line is equal to the instantaneous acceleration.
•
For any straight section, the displacement can be calculated by using
#v + v&
Δx = % 0
(t . Use the velocity from
$ 2 '
the beginning of the section (vo), the velocity at the end of the section (v), and the amount of time that passed
during the interval (t = tfinal - tinitial).
•
Using the above equation is equivalent to finding the area€between the section and the x axis – this is another
way of thinking of how to find the displacement in a section.
•
The displacement during any straight section is equal to the area “under the line”.
•
In summary: Initial and final velocities and times can be read directly from graph. Acceleration is
equal to slope of straight section. Displacement during any straight section is equal to the area “under
the line”.
Position (m)
10
8
6
4
2
0
0
5 10 15 20 25 30 35 40 45
Time (s)
Example Questions:
1) What was the velocity from t = 15 to t = 20 s?
2) What was the displacement during the first 35 s?
answer: v = Δx/Δt = (6 m – 4 m)/(20 s – 15s) = 0.4 m/s
answer: Δx = x – xo = 2 m – 0 m = 2 m
Velocity (m/s)
10
8
6
4
2
0
0
5 10 15 20 25 30 35 40 45
Time (s)
Example Questions:
1) What was the acceleration from t = 15 to t = 20 s? answer: a = Δv/Δt = (6 m/s – 4 m/s)/(20 s – 15s) = 0.4 m/s2
#
m&
%0 + 4 s (
# v0 + v &
answer: Δx = %
(t = Δx = %
(5s = 10 m
$ 2 '
% 2 (
$
'
2) What was the displacement during the first 5 s?
To find the displacement for the first 20 s you need to calculate it for each of the three straight intervals (t = 0 to t = 5s; t = 5 to
15 s; t = 15 to 20s) and add them up.
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