A short powerpoint presentation explaining some techniques to analyzing position-time plots and its derivatives without having to compute any numbers.

1. Analyzing Position-Time
Plots
Phys 101 Learning Object
Nicholas Najy

2. To the left is a graph of an object
undergoing simple harmonic motion
(SMH). In this presentation we will look at
methods for interpreting these graphs,
using background knowledge in kinematics
and basic ideas of integral and differential
calculus.
There are several important variables to
define:
A = The amplitude of the wave function
T = The period of one oscillation
ƒ = The frequency of the wave function
t = The time at a given moment
x(t) = The position at a time “t”
x’(t) = v(t) = The velocity at time “t”
x’’(t) = v’(t) = a(t) = The acceleration at
time “t”

3. t
x(t)
A
B
Interpreting Position-Time Plots
Consider point A. Here we are at one of the points of maximum positive position
for the object. Since velocity is just the derivative of the position as a function of
time, we know that the velocity of the object at this point is zero, because the
slope of the tangent line to point A is zero.
If we look at point C, we can see that the velocity is also zero, but this time the
object is at its maximum negative velocity. Therefore, we can conclude that the
zeroes of an object’s velocity occur at the points of maximum amplitude.
If we consider point B, we see now that the position of the object is zero.
However, the tangent at this point has the steepest slope. Point D has an equal
and opposite slope. Therefore, the points of greatest velocity occur at the points
of zero amplitude.
Another interesting pattern is that we can see that the slope of the plot
consistently decreases above the x-axis, and consistently increases below the x-
axis. When we look at second derivative of the function, we can see that this
corresponds with where the x’’(t) is positive (concave positive), and where it is
negative (concave down). Thus we can visually see that an object undergoing
SMH is always accelerating back to the equilibrium point.
Key Principles:
1. Points of maximum amplitude on a position plot indicate zero velocity
2. Points of zero amplitude on a position plot indicate maximum velocity
3. Downward concavity indicates negative acceleration
4. Upward concavity indicates positive acceleration
C
D

4. t
x’(t)
H
E
Interpreting First Derivative Position-Time Plots
Now we’ll look at the velocity-time plot of the same object’s motion. To interpret this graph
we will need to remember some of the principles from the last slide. At points F and H, we
have the maximum negative and positive velocities, which means at these points the position
is zero. However from the tangent lines at these points we can also see that the acceleration is
zero. In fact, at every zero for position, there is also a zero for acceleration.
From integration, we know that the area under a velocity-time plot over a given interval is the
displacement of the object over that interval. When looking at areas above and below the x-
axis, the areas below are subtracted from those above, to get a net area. Thus, if we wanted to
find the maximum possible displacement of any interval, it would have to be where the
function is either always positive or negative. In the interval [E,G], the velocity is always
negative, which means the object only ever moved in the negative direction. Therefore, the
maximum displacement of the object can be shown as:
𝐼 =
𝐸
𝐺
𝑥′
𝑡 𝑑𝑡 = [𝑥 𝑡 ]
Where E and G simply indicate points before and after a half wave cycle that is either all
above or below the x-axis.
Key Principles:
1. Points of maximum amplitude on a velocity plot indicate zero acceleration
2. Points of zero amplitude on a velocity plot indicate maximum acceleration
3. The area under a half wave cycle that is either entirely above or below the x-axis indicates
the maximum displacement of the object (not from the equilibrium point, but from one point
to another)
F
G
I1
𝐸
𝐺

5. t
x’’(t)
I
J
Interpreting Second Derivative Position-Time Plots
Now we’ll look at the acceleration-time plot of the same object’s motion. If we look
at points I and K, we can see that the slope is zero and we have maximum positive
and negative acceleration. However, another interesting feature to note is that these
points also correspond with points A and C on the position-time graph. We can
observe that at every point of maximum positive or negative position, there will be
zero velocity and maximum negative or positive acceleration.
As mentioned before, the positive and negative sections of the plot correspond with
the concavity of the position-time plot for the object’s motion. In fact, the value of the
function for the acceleration of an object will always be opposite of the value of the
function for the position, so the acceleration will always be negative when the
position is positive, and vice versa.
The area under an acceleration plot over a given interval indicates the change in
velocity over that given interval. Thus the maximum possible change in velocity of an
interval can be shown as:
𝐼2 =
𝐽
𝐿
𝑥′′
𝑡 𝑑𝑡 = [𝑥′ 𝑡 ]
Where J and L simply indicate points before and after a half wave cycle that is either
all above or below the x-axis.
Key Principles:
1. Points of maximum amplitude on an acceleration plot indicate zero velocity and
maximum position
2. Points of zero amplitude on an acceleration plot indicate maximum velocity and
zero position
3. The area under a half wave cycle that is either entirely above or below the x-axis
indicates the maximum change in velocity of the object
K
L
I2
𝐽
𝐿

6. Now, using what we studied in these few slides, you can hopefully
identify the points of maximum positive and negative position,
velocity, and acceleration, and intervals of maximum displacement
and change in velocity, for an object undergoing SMH, simply by
looking at a position-time plot, or even a first or second derivative
position-time plot.
For practice, use the methods we discussed to analyze the graph
below. Try answering the questions on the right.
1. Identify the points of maximum positive
acceleration.
2. Identify an interval of maximum decrease
in velocity.
3. Identify the intervals of upwards concavity
for the derivative of this function. (Hint:
This is an extended application of what was
explained in slide 2)