### Representing Motion

```Representing Motion
Chapter 2 (pg 30-55)
Do Now
• Why is it important to describe and
analyze motion?
•
•
•
•
How fast?
How far?
Slowing/Speeding?
Rest/Constant velocity
• What are some of the different types of
motion?
• Translational – motion along a straight line
• Circular – motion along a circular path
• Rotational – Rotation about a fixed point
Chapter Objectives
●Represent motion through the use of
words, motion diagrams, and graphs.
●Use the terms position, distance,
displacement, and time interval in a
scientific manner to describe motion.
Motion
• Motion is instinctive
• Eyes will notice moving objects more readily than stationary ones
• Object changes position
• Motion can occur in many directions and paths
(2.1)Picturing Motion
(2.2) Where and When?
• Analyze motion diagrams to describe motion.
• Develop a particle model to represent a moving object.
• Define coordinate systems for motion problems and recognize that it
affects the sign of an object’s position.
• Define displacement.
position or displacement.
Picturing Motion
• A description of motion relates to a PLACE
and TIME.
• Answers the questions WHERE? and
WHEN?
TIME
PLACE
Motion Diagrams
• Series of images showing the
positions of a moving object at equal
time intervals.
Particle Model
• Simplified version of a
motion diagram in
which the object in
motion is replaced by
a series of single
points.
• Size of object much less
than the distance moved
• Internal motions of object
ignored
Particle Model
Motion Diagram & Particle Model
Practice
• Draw a particle model for the
motion diagram above of a car
coming to a stop.
Describe the motion of the
bird…
Draw a particle model….
Use the particle model to draw motion
diagrams for two runners in a race.
When the first runner crosses the
finish line, the second runner is ¾ of
the way to the finish line.
• First Runner
• Second Runner
Must have same
number of particles
to represent equal
time.
How are the two particle models
different? Describe the motion of each.
A.
B.
A. Eight time intervals & Constant
velocity (equal spacing)
B. Five time Intervals & speeding up
(spacing is getting farther)
Which statement describes best the
motion diagram of an object in motion?
A.
a graph of the time data on a horizontal axis and the
position on a vertical axis
B.
a series of images showing the positions of a moving
object at equal time intervals
C.
a diagram in which the object in motion is replaced
by a series of single points
D.
a diagram that tells us the location of the zero point
of the object in motion and the direction in which the
object is moving
What is the purpose of drawing a
motion diagram or a particle model?
A. to calculate the speed of the object in
motion
B. to calculate the distance covered by the
object in a particular time
C.
to check whether an object is in motion
D. to calculate the instantaneous velocity of
the object in motion
Coordinate System
• Tells you the location of the zero point of the
variable you are studying and the direction in
which the values of the variable increase.
• ORIGIN
• The point at which both variables have the
value zero
Coordinate System
• Motion is RELATIVE
• You can define a coordinate system any way you want, but some
are more useful than others.
Coordinate systems
Axis of the
coordinate system
• This coordinate system works as well but is
not as convenient to use as the first one.
• Try to always pick your origin where
motion begins.
Position & Distance
• You can indicate how far an object is from the
origin by drawing an arrow from the origin to the
point representing the object.
• The two arrows indicate the runner’s POSITION
at two different times. (vector)
• Separation between an object and the origin
• The length of how far an object is from the origin
indicates DISTANCE. (scalar)
Refer to the figure and calculate the distance
between the two signals?
A. 3 m
C. 5 m
B. 8 m
D. 5 cm
Vectors & Scalars
• SCALARS: quantities that are just numbers without any
direction
• Magnitude (number) only
• Examples: time, volume, mass,
temperature
• VECTORS: quantities that have both magnitude (size) and
direction
• Represented by arrows
• Examples: velocity, acceleration, force, momentum
Tail
Tip
• Just like you can add scalars, you can also add vectors.
• Algebraically (look at later) & Graphically
• Place vectors tail to tip
• Place the tail of the 2nd vector next to the tip of the 1st vector
• RESULTANT - the vector that represents the sum of two or more
other vectors
• is drawn from the tail of the first vector to the tip of the last vector.
Vector 1
Vector 3
Resultant
Vector 3
Vector 1
Vector 1
Vector 3
Like scalars, vectors can be added in different
order and still have the same resultant.
Distance vs. Displacement
• DISTANCE
• Actual length traveled
• Scalar measurement
• Path dependent
• DISPLACEMENT
• Change in position
• Vector measurement
• Path independent
∆x = xf – xi
Distance vs. Displacement
• Find distance and
displacement for
the following races:
Race
100 m
400 m
1 mile
Distance
Displacement
Practice
Ex: Jared walks 75 m down the block heading
east when he realizes he dropped his book. He
turns around and walks 15 m until he finds his
book.
• Draw vectors to represent Jared’s motion.
• Find the distance that Jarred walked.
• Find Jared’s displacement.
What is displacement?
• A. the vector drawn from the initial position to
the final position of the motion in a coordinate
system
• B. the distance between the initial position and
the final position of the motion in a coordinate
system
• C. the amount by which the object is displaced
from the initial position
• D. the amount by which the object moved from
the initial position
Ch. 2.3 Objectives
• Develop position-time graphs for moving
objects.
• Use position-time graphs to interpret an
object’s position or displacement.
• Make motion diagrams, pictorial
representations, and position-time graphs
that are equivalent in describing an
object’s motion.
Graphs
• Named as y-axis vs. x-axis
• Also as Dependent vs. Independent
• Ex. Position vs. time
• Place position on the y-axis and time on the x-axis
Always play close attention to
the units.
Units are key to analyzing
graphs…
Analyzing Graphs
Slope:
Look at the units of the slope to
see if it corresponds to a
measurement.
Area:
look at the units for the area under
the curve to see if it corresponds to
a measurement.
Position-Time Graphs
SLOPE: Rise/Run
m=
∆
∆
=
∆
∆
=

Velocity
Where?
&
When?
y-axis
Dependent
Variable
x-axis:
Independent
variable
From the following position-time graph of two
brothers running a 100-m dash, at what time do both
brothers have the same position? At what position?
Position-Time Graphs
• When does runner B pass runner A?
• 45 seconds into the race
• Where does runner B pass runner A?
• 190 m
Position-Time Graphs
• What is the displacement of the runner between 5 s
and 10 s?
10 m
What is happening in this
graph?
What is happening in each?
A.
C.
B.
D.
Position-Time Graph - SLOPE
• Analyze the units
• Slope = rise over run
• m = ∆y / ∆x
• Slope = m / s
• m/s is the unit for velocity
• The slope of a position-time graph is the average
velocity
Position-Time Graph - AREA
• Analyze units only
• Area under the curve
• Area of a triangle A = ½ b * h
• ½ is a constant and has no units
• Base has units of time (s)
• Height has units of position (m)
• Area = (m)(s)
• We do not have any measurements that have the units
(s)(m): thus the area of a position-time graph does not
have any meaning.
Ch. 2.4 Objectives
• Define Velocity
• Differentiate between speed and velocity
• Create pictorial, physical, and mathematical
models of motion problems
Average Velocity
• Defined as the change in position, divided by the
time during which the change occurred.
• How fast in a given direction?
• Vector quantity
• Same direction as the displacement (Δx)

−
∆
=
=
∆
−
Average Speed
• The absolute value of the slope of a
position-time graph.
• It describes how fast the object is moving.

=

Instantaneous Velocity
• Velocity at a given instant
• Slope of the line drawn on the x-t graph at the given
instant
• Need calculus to find unless object moving at a
constant velocity
Velocity vs. Speed
• Speed is the distance
divided by the time
during which the
distance was
traveled.
• Scalar (No direction)
Race
Distance (m)
Displacement
(m)
Time (s)
100 m
20
400 m
80
1 mile
400
Speed (m/s)
Velocity (m/s)
Position-Time Graphs - Velocity
Suppose you recorded two joggers in one motion diagram, as
shown in the figure above. From one frame to the next, you can
see that the position of the jogger in red shorts changes more
than that of the one wearing blue.
What would the x-t graph look like if they both started at the
same time?
Position-Time Graphs
• You can create a position-time graph if you know the
position and time of the joggers at different points.
• Need a minimum of two data points in order to
create a x-t graph.
Position-Time Graphs (velocity)
• Can find the velocity of each jogger by calculating the
slope of the line.
• Red Jogger
• v = m = (6m – 2 m)/ (3s – 1s)
• v = 2 m/s
• Blue Jogger
• v = m = (3m – 2m) / (3s – 2s)
• v = 1 m/s
What is happening?
1 - Compare the
velocities for
each of the
segments.
2- Rate the
segments in
increasing speed.
```