A Mathematical
Model of Motion
CHAPTER 5
PHYSICS
5.1 Graphing Motion in One
Dimension
Interpret graphs of position versus time for
a moving object to determine the velocity
of the object
 Describe in words the information
presented in graphs and draw graphs from
descriptions of motion
 Write equations that describe the position
of an object moving at constant velocity

Parts of a
Graph
 X-axis
 Y-axis
 All
axes must be labeled with
appropriate units, and values.
5.1 Position vs. Time
The x-axis is always
“time”
 The y-axis is always
“position”
 The slope of the line
indicates the velocity
of the object.
 Slope = (y2-y1)/(x2-x1)

 d1-d0/t1-t0
 Δd/Δt
Position vs. Time
20
18
16
Position (m)
14
12
10
8
6
4
2
0
1
2
3
4
5
6
Time (s)
7
8
9
10
Uniform Motion
 Uniform
motion is defined as equal
displacements occurring during
successive equal time periods
(sometimes called constant velocity)
 Straight lines on position-time graphs
mean uniform motion.
Given below is a diagram of a ball rolling along a table. Strobe
pictures reveal the position of the object at regular intervals of time,
in this case, once each 0.1 seconds.
Notice that the ball covers an equal distance between flashes. Let's assume this
distance equals 20 cm and display the ball's behavior on a graph plotting its xposition versus time.
The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This
represents the ball's average velocity as it moves across the table. Since the
ball is moving in a positive direction its velocity is positive. That is, the ball's
velocity is a vector quantity possessing both magnitude (200 cm/sec) and
direction (positive).
Steepness of slope on PositionTime graph
Slope
is related to velocity
Steep slope = higher
velocity
Shallow slope = less
velocity
Different Position. Vs. Time graphs
Position vs. Time
Uniform Motion
Position (m)
20
Accelerated
Motion
15
10
5
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Constant positive velocity
(zero acceleration)
Increasing positive velocity
(positive acceleration)
Position vs. Time
Position (m)
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Constant negative velocity
(zero acceleration)
Decreasing negative velocity
(positive acceleration)
Different Position. Vs. Time
Changing slope means changing velocity!!!!!!
Decreasing negative slope = ??
Increasing negative slope = ??
X
B
A
t
C
A … Starts at home (origin) and goes forward
slowly
B … Not moving (position remains constant as time
progresses)
C … Turns around and goes in the other direction
quickly,
passing up home
During which intervals was he traveling in a positive direction?
During which intervals was he traveling in a negative direction?
During which interval was he resting in a negative location?
During which interval was he resting in a positive location?
During which two intervals did he travel at the same speed?
A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec
x
B
C
Graphing w/
Acceleration
t
A
D
A … Start from rest south of home; increase speed gradually
B … Pass home; gradually slow to a stop (still moving north)
C … Turn around; gradually speed back up again heading south
D … Continue heading south; gradually slow to a stop near the
starting point
You try it…..
Using the Position-time graph given to
you, write a one or two paragraph “story”
that describes the motion illustrated.
 You need to describe the specific motion
for each of the steps (a-f)
 You will be graded upon your ability to
correctly describe the motion for each
step.

Tangent
Lines
x
t
On a position vs. time graph:
SLOPE
VELOCITY
SLOPE
SPEED
Positive
Positive
Steep
Fast
Negative
Negative
Gentle
Slow
Zero
Zero
Flat
Zero
Increasing &
Decreasing
x
t
Increasing
Decreasing
On a position vs. time graph:
Increasing means moving forward (positive direction).
Decreasing means moving backwards (negative direction).
x
Concavity
t
On a position vs. time graph:
Concave up means positive acceleration.
Concave down means negative acceleration.
x
Q
R
P
Special
Points
S
Inflection Pt.
P, R
Peak or
Valley
Q
Time Axis
Intercept
P, S
Change of concavity,
change of acceleration
Turning point, change of
positive velocity to
negative
Times when you are at
“home”, or at origin
t
5.2 Graphing Velocity in One
Dimension
Determine, from a graph of velocity versus
time, the velocity of an object at a specific
time
 Interpret a v-t graph to find the time at
which an object has a specific velocity
 Calculate the displacement of an object
from the area under a v-t graph

5.2
Velocity vs. Time
X-axis is the
“time”
 Y-axis is the
“velocity”
 Slope of the
line = the
acceleration
Velocity vs. Time

20
18
16
Velcoity (m/s)
14
12
10
8
6
4
2
0
1
2
3
4
5
6
Time (s)
7
8
9
10
Different Velocity-time graphs
Different Velocity-time graphs
Velocity vs. Time
Velocity (m/s)
20
15
10
5
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
Time (s)
Velocity vs. Time
Velocity (m/s)
25
20
15
10
5
0
1
2
3
4
5
6
Time (s)
Velocity vs. Time
Horizontal lines = constant velocity
 Sloped line = changing velocity

 Steeper
= greater change in velocity per
second
 Negative slope = deceleration
Acceleration vs. Time
Acceleration vs. Time
12
10
Acceleration (m/s^2)
Time is on the x-axis
 Acceleration is on
the y-axis
 Shows how
acceleration
changes over a
period of time.
 Often a horizontal
line.
8
6
4
2
0
1
2
3
4
5
6
Time (s)
7
8
9
10
x
All 3 Graphs
t
v
t
a
t
Real life
Note how the v graph is pointy and the a graph skips. In real life,
the blue points would be smooth curves and the orange segments
would be connected. In our class, however, we’ll only deal with
constant acceleration.
v
t
a
t
Constant Rightward Velocity
Constant Leftward Velocity
Constant Rightward
Acceleration
Constant Leftward Acceleration
Leftward Velocity with
Rightward Acceleration
Graph Practice
Try making all three graphs for the following scenario:
1. Newberry starts out north of home. At time zero he’s
driving a cement mixer south very fast at a constant speed.
2. He accidentally runs over an innocent moose crossing
the road, so he slows to a stop to check on the poor moose.
3. He pauses for a while until he determines the moose is
squashed flat and deader than a doornail.
4. Fleeing the scene of the crime, Newberry takes off again
in the same direction, speeding up quickly.
5. When his conscience gets the better of him, he slows,
turns around, and returns to the crash site.
Area Underneath v-t Graph


If you calculate the area underneath
a v-t graph, you would multiply
height X width.
Because height is actually velocity
and width is actually time, area
underneath the graph is equal to
 Velocity X time or
V·t
Remember that Velocity = Δd
Δt
 Rearranging, we get Δd = velocity X Δt


So….the area underneath a velocity-time
graph is equal to the displacement during
that time period.
v
“positive area”
Area
t
“negative area”
Note that, here, the areas are about equal, so even though a
significant distance may have been covered, the displacement is
about zero, meaning the stopping point was near the starting point.
The position graph shows this as well.
x
t
Velocity vs. Time




The area under a velocity time graph indicates
the displacement during that time period.
Remember that the slope of the line indicates
the acceleration.
The smaller the time units the more
“instantaneous” is the acceleration at that
particular time.
If velocity is not uniform, or is changing, the
acceleration will be changing, and cannot be
determined “for an instant”, so you can only find
average acceleration
5.3 Acceleration
Determine from the curves on a velocitytime graph both the constant and
instantaneous acceleration
 Determine the sign of acceleration using a
v-t graph and a motion diagram
 Calculate the velocity and the
displacement of an object undergoing
constant acceleration

5.3 Acceleration


Like speed or velocity,
acceleration is a rate
of change, defined as
the rate of change of
velocity
Average Acceleration
= change in velocity
Elapsed time
V 1 V 0
a
t
Units of acceleration?
Rearrangement of the equation
V 1 V 0
a
t
at  v1  v 0
v 0  at  v1
v1  v0  at
Finally…
v1  v0  at

This equation is to be used to find (final)
velocity of an accelerating object. You can
use it if there is or is not a beginning
velocity
Displacement under Constant
Acceleration

Remember that displacement under
constant velocity was
Δd = vt
With
or
d1 = d0 + vt
acceleration, there is no one
single instantaneous v to use, but
we could use an average velocity
of an accelerating object.
Average velocity of an accelerating
object would simply be ½ of sum of
beginning and ending velocities
Average velocity of an accelerating object
V = ½ (v0 + v1)
So…….
d1  d 0  vt
d1  d 0  1 / 2(v1  v0)t
d1  d 0  1 / 2(v1  v0)t
Key equation
Some other equations
d 1  d 0  v 0t  1 / 2at
2
This equation is to be used to find
final position when there is an
initial velocity, but velocity at time
t1 is not known.
If no time is known, use this to find
final position….
v1  v 0
d1  d 0 
2a
2
2
Finding final velocity if no time is
known…
v1  v0  2a(d 1  d 0)
2
2
The equations of importance
V 1 V 0
a
t
v1  v0  at
d  d 0  1 / 2(v1  v0)t
d  d 0  v 0t  1 / 2at
v1  v 0
d1  d 0 
2a
2
2
2
v1  v0  2a(d 1  d 0)
2
2
Practical Application
Velocity/Position/Time equations







Calculation of arrival times/schedules of aircraft/trains
(including vectors)
GPS technology (arrival time of signal/distance to
satellite)
Military targeting/delivery
Calculation of Mass movement (glaciers/faults)
Ultrasound (speed of sound) (babies/concrete/metals)
Sonar (Sound Navigation and Ranging)
Auto accident reconstruction
Explosives (rate of burn/expansion rates/timing with det.
cord)
5.4 Free Fall
Recognize the meaning of the acceleration
due to gravity
 Define the magnitude of the acceleration
due to gravity as a positive quantity and
determine the sign of the acceleration
relative to the chosen coordinate system
 Use the motion equations to solve
problems involving freely falling objects

Freefall
 Defined
as the motion of an
object if the only force acting
on it is gravity.
 No
friction, no air resistance, no drag
Acceleration Due to Gravity


Galileo Galilei recognized
about 400 years ago that,
to understand the motion
of falling objects, the
effects of air or water
would have to be ignored.
As a result, we will
investigate falling, but
only as a result of one
force, gravity.
Galileo Galilei 1564-1642
Galileo’s Ramps

Because gravity causes
objects to move very fast,
and because the timekeepers available to
Galileo were limited,
Galileo used ramps with
moveable bells to “slow
down” falling objects for
accurate timing.
Galileo’s Ramps
Galileo’s Ramps

To keep “accurate” time, Galileo used a water
clock.

For the measurement of time, he employed a large
vessel of water placed in an elevated position; to the
bottom of this vessel was soldered a pipe of small
diameter giving a thin jet of water, which he collected
in a small glass during the time of each descent... the
water thus collected was weighed, after each
descent, on a very accurate balance; the difference
and ratios of these weights gave us the differences
and ratios of the times...
Displacements of Falling Objects
Looking at his results, Galileo realized that
a falling ( or rolling downhill) object would
have displacements that increased as a
function of the square of the time, or t2
 Another way to look at it, the velocity of an
object increased as a function of the
square of time, multiplied by some
constant.

 Galileo
also found that all objects,
no matter what slope of ramp he
rolled them down, and as long as
the ramps were all the same
height, would reach the bottom
with the same velocity.
Galileo’s Finding
 Galileo
found that, neglecting
friction, all freely falling objects
have the same acceleration.
Hippo & Ping Pong Ball
In a vacuum, all bodies fall at the same rate.
If a hippo and a
ping pong ball were
dropped from a
helicopter in a
vacuum (assuming
the copter could fly
without air), they’d
land at the same
time.
When there’s no air resistance, size and shape matter not!
Proving Galileo Correct
Galileo could not
produce a vacuum
to prove his ideas.
That came later with
the invention of a
vacuum machine,
and the
demonstration with
a guinea feather
and gold coin
dropped in a
vacuum.
Guinea Feather and Coin/NASA
demonstrations
Acceleration Due to Gravity

Galileo calculated that all freely falling
objects accelerate at a rate of
9.8
2
m/s
This value, as an acceleration, is known as g
Acceleration Due to Gravity
 Because
this value is an acceleration
value, it can be used to calculate
displacements or velocities using the
acceleration equations learned
earlier. Just substitute g for the a
Example problem

A brick is dropped from a high building.
 What
is it’s velocity after 4.0 sec.?
 How far does the brick fall during this time?
The Church’s opposition to new
thought




Church leaders of the time held the same views
as Aristotle, the great philosopher.
Aristotle thought that objects of different mass
would fall at different rates…makes sense
huh??????
All objects have their “natural position”. Rocks
fall faster than feathers, so it only made sense
(to him)
Galileo, in attempting to convince church leaders
that the “natural position” view was incorrect,
considered two rocks of different mass.
Falling Rock Conundrum
Galileo presented this in his book
Dialogue Concerning the Two Chief
World Systems(1632) as a discussion
between Simplicio (as played by a church
leader) and Salviati (as played by Galileo)
 Two rocks of different masses are dropped
 Massive rock falls faster???

Rocks continued




Now consider the two rocks held together by a
length of string.
If the smaller rock were to fall slower, it would
impede the rate at which both rocks would fall.
But the two rocks together would actually have
more mass, and should therefore fall faster.
A conundrum????? The previously held views
could not have been correct.
Galileo had data which proved Aristotle
and the church wrong, but church leaders
were hardly moved in their position that all
objects have their “correct position in the
world”
 Furthermore, the use of Simplicio (or
simpleton) as the head of the church in his
dialog, was a direct insult to the church
leaders themselves.
