Name: _____________________________________________
Period: _____
Physics
Unit I: Motion
Subunit A: Constant Velocity
Motion Maps, graphs and equations can describe and predict motion.
Equations
Variables, Units
NOTES:
Unit I-A Objectives
What you should know when all is said and done
1. You should distinguish between a scalar and a vector:
a. know the difference between distance and displacement.
b. know the difference between speed and velocity.
c. know the difference between average and instantaneous speed and velocity.
2. You should be able to determine the average velocity of an object in two ways:
a. determining the slope of an x vs t graph.
b. using the equation v = Δx/Δt
3. You should be able to determine the displacement of an object in two ways:
a. finding the area under a v vs t graph.
b. using the equation Δx = vt
4. Given a motion map, you should be able to:
a. describe the motion of the object (starting position, direction of motion, velocity)
b. draw the corresponding x vs t graph
5. Given an x vs t graph, you should be able to:
a. describe the motion of the object (starting position, direction of motion, velocity)
b. draw the corresponding v vs t graph
c. draw a motion map for the object.
d. determine the average velocity of the object (slope).
e. write the mathematical model which describes the motion.
6. Given a v vs t graph, you should be able to:
a. describe the motion of the object (direction of motion, how fast)
b. draw the corresponding x vs t graph
c. determine the displacement of the object (area under curve).
d. draw a motion map for the object.
e. write a mathematical model to describe the motion.
Unit I-A: Constant Velocity
Worksheet 1
1. When choosing a 1-dimensional horizontal reference frame, or coordinate system, we
usually choose the positive direction to be toward the (right / left) and the negative direction to
be toward the (right / left).
2. ______ A quantity that only describes how much (magnitude) is referred to as a
A) scalar quantity
B) vector quantity
3. ______ A quantity that describes how much (magnitude) and which way (direction) is referred
to as a
A) scalar quantity
B) vector quantity
3. Some examples of scalars are:
4. Some examples of vectors are:
5. To measure displacement, or change in position Δx,
A) measure every meter moved
B) take the final position minus the initial position only
6. True or False: An object can be moving for 10 seconds and still have zero displacement.
7. True or False: It is possible for an object to move for 10 seconds at a high speed and end up
with negative displacement.
8. Johnny drives to 1920 miles in 32 hours and returns home by the same route in the same
amount of time.
A) Determine his average speed.
B) Determine his average velocity.
C) Compare these two values and explain any differences.
9.
This person has a displacement of ________ km.
a. 0 km
b. 3 km
c. 3 km, E
d. 3 km, W
e. 5 km
f. 5 km, N g. 5 km, S
h. 6 km
i. 6 km, E j. 6 km, W k. 31 km
l. 31 km, E
m. 31 km, W
n. None of these.
9. AAcross-country
skiermoves
movesfrom
from
location
A location
to location
to location
C to location
D. leg
Each
leg
10.
cross-country skier
location
A to
B toBlocation
C to location
D. Each
of the
of the
back-and-forth
motion
takes
1
minute
to
complete;
the
total
time
is
3
minutes.
back-and-forth motion takes 1 minute to complete; the total time is 3 minutes.
A) What
is the
distance
skierduring
duringthe
thethree
three
minutes
of recreation?
a. What
is the
distancetraveled
traveled by
by the
the skier
minutes
of recreation?
b. What is the net displacement of the skier during the three minutes of recreation?
B) What is the net displacement of the skier during the three minutes of recreation?
c. What is the displacement during the second minute (from 1 min. to 2 min.)?
C) What is the displacement during the second minute (from 1 min. to 2 min.)?
d. What is the displacement during the third minute (from 2 min. to 3 min.)?
D) What is the displacement during the third minute (from 2 min. to 3 min.)?
E) Calculate the average speed (in m/min) and the average velocity (in m/min) of the skier
during the three minutes of recreation.
© The Physics Classroom, 2009
10.
Page 2
Position (m)
10
A) Equation: _______________________
B) This object starts ______ meters to the (right / left) of
the origin and travels with ______________
______________ in a straight line in the (+ / - ) direction
for 10 seconds at a speed of ____________.
Time (s)
0
0
10
11.
Position (m)
A) Equation:
10
B) This object starts ______ meters to the (right / left)
of the origin and travels with ______________
______________ in a straight line in the (+ / - )
direction for 10 seconds at a speed of
____________.
Time (s)
0
0
10
12.
Position (m)
0
Time (s)
A) Equation:
B) Describe the motion of this object as
above.
-10
0
10
UNIT II READING: MOTION MAPS
A motion map represents the position, velocity, and acceleration of an object at various clock
readings. (At this stage of the class, you will be representing position and velocity only.)
Unit I-A Reading: Motion Maps & Graphs
Suppose that you took a stroboscopic picture of a car moving to the right at constant velocity
A motion
mapimage
represents
thethe
position,
acceleration
of an object at various clock
where each
revealed
positionvelocity,
of the carand
at one-second
intervals.
readings. (At this stage of the class, you will be representing position and velocity only.)
Suppose that you took a stroboscopic picture of a car moving to the right at constant velocity
where each image revealed the position of the car at one-second intervals.
This is the motion map that represents the car. We model the position of the object with a small
point. At each position, the object’s velocity is represented by a vector.
This is the motion map that represents the car. We model the position of the object with a small
point. At each position, the object’s velocity is represented by a vector.
If the car were traveling at greater velocity, the strobe photo might look like this:
If the car were traveling at greater velocity, the strobe photo might look like this:
The corresponding motion map has the points spaced farther apart, and the velocity vectors are
longer, implying that the car is moving faster.
The corresponding motion map has the points spaced farther apart, and the velocity vectors are
longer, implying that the car is moving faster.
If the car were moving to the left at constant velocity, the photo and motion map might look like
If the
this:car were moving to the left at constant velocity, the photo and motion map might look like
this:
More complicated motion can be represented as well.
More complicated motion can be represented as well.
Here, an object moves to the right at constant velocity, stops and remains in place for two
seconds, then moves to the left at a slower constant velocity.
Here, an object moves to the right at constant velocity, stops and remains in place for two
seconds, then moves to the left at a slower constant velocity.
'Modeling Workshop Project 2002
1
Unit II Reading-Motion Maps v2.0
Consider the interpretation of the motion map below. At time t = 0, cyclist A starts moving to
Consider
interpretation
motion
maptobelow.
Atoftime
= 0, cyclist A starts moving to
the
right atthe
constant
velocity,of
atthe
some
position
the right
the torigin.
the rightthe
at constant
velocity,
at some
position
to the At
right
of the
origin.
Consider
interpretation
of the
motion
map below.
time
t = 0,
cyclist A starts moving to
the right at constant velocity, at some position to the right of the origin.
Cyclist B starts at the origin and travels to the right at a constant, though greater velocity.
Cyclist
B
starts
the origin
andboth
travels
the same
right at
constant,
greater
velocity.
At t =B
3 starts
s, B overtakes
A (i.e.,
have
the
but though
Bthough
is moving
faster).
Cyclist
atatthe
origin
and
travels
totothe
right
atposition,
aaconstant,
greater
velocity.
At t = 3 s, B overtakes A (i.e., both have the same position, but B is moving faster).
At t = 3 s, B overtakes A (i.e., both have the same position, but B is moving faster).
graphicalrepresentation
representationof
ofthe
thebehavior
behavior of
ofcyclists
cyclistsAAand
andBBwould
wouldlike
likethis:
this:
AAgraphical
A graphical representation of the behavior of cyclists A and B would like this:
You could also represent the behavior algebraically as follows:
You could
could also
alsorepresent
representthe
thebehavior
behavior
algebraically
follows:
You
algebraically
asas
follows:
x = v At + x0 , for A
= vvAtt +
xx =
+ xx0,, forfor
A A where vB > vA
x x= =vBAvt,Bt, 0 for
forBBwhere v B > vA
x = vBt,
for B
Throughout
semester, you will be representing the behavior of objects in motion in multiple
where
vB > vthis
A
Throughout
this semester,
you will
be representing
the behavior
of objects in motion in multiple
ways: diagramatically
(motion
maps),
graphically and
algebraically.
ways:
diagramatically
(motion
andthe
algebraically.
Throughout
this semester,
youmaps),
will begraphically
representing
behavior of objects in motion in multiple
ways: diagramatically (motion maps), graphically, and algebraically.
Hints on drawing your own motion maps:
1. Draw dots indicating the position of the object at equal time intervals, i.e. each second.
2. Attach arrows to the dots indicating the direction of motion. Make the arrow length half of the
space between the dots to make your motion map easy to read.
3. When an object is stopped for several time intervals, draw multiple dots at the same position.
4. Make sure your sequence of arrows has a logical flow so that the motion is clearly
communicated.
Unit I-A: Constant Velocity
Worksheet 2
1. Consider the position vs. time graph below for cyclists A and B.
A) Do the cyclists start at the same point? How
do you know? If not, which is ahead?
x
(m)
A
B
B) At t = 7 s, which cyclist is ahead? How do
you know?
C) Which cyclist is traveling faster at t = 3 s?
How do you know?
5
t (s)
D) Are their velocities equal at any time? How
do you know?
E) What is happening at the intersection of line A and B?
F) Draw a motion map for cyclists A and B.
2. Consider the position vs. time graph below for cyclists A and B.
A) How does the motion of the two cyclists in
this graph compare to the previous question?
B) Which cyclist has the greater speed?
How do you know?
D) Which cyclist has traveled further during the first 5 seconds? How do you know?
F) Draw a motion map for cyclists A and B.
3. A) Using the Data Table below, create a graph and write the corresponding equation
Time
(s)
0
1
2
3
4
5
Position
(m)
15
12
9
6
3
0
B) Find the equation of the line.
C) Describe the motion of the object.
D) Draw a motion map for the object.
4. The graph at below shows the motion of a girl on a jet ski moving in a straight line.
A) What is the total distance she travels?
B) What is her total displacement?
C) What is her average speed?
D) What is her average velocity?
Unit I-A: Constant Velocity
Worksheet 3
1. A motorized scooter was observed to be at the following positions at the times listed below:
t (s)
x (m)
0
11
1
9
2
7
A) Draw a motion map for
3
5
the scooter.
4
3
5
1
B) Plot the position vs. time
graph for the scooter.
C) Was the scooter’s velocity constant throughout
the
whole interval? How do you know?
D) What is the velocity of the scooter? How do you
know?
E) What would his position be at t = 7 s if his velocity remains constant? Use your equation to
find out.
2. Robin, roller-skating down a sidewalk, was observed to be at the following positions at the
times listed below:
t (s)
x (m)
0
2
A) Plot a position vs. time graph for the
1
5
skater.
2
8
5
14
8
20
10
26
B) Find the velocity of the skater.
C) What is the equation that models her motion?
D) How far from the origin was she at t = 6 s? How do you
know?
E) What would be her position at t = 12 seconds if her velocity
remains constant?
3. The following data were obtained for the skater’s second trial:
t (s)
0
2
4
6
8
10
x (m)
4
10
16
22
28
34
A) Plot the position vs. time graph for the
skater.
B) Find the velocity of the skater.
C) What is the equation that models her motion?
D) How far from the origin was she at t = 5 s? How do you know?
E) What would be her position at t = 12 seconds if her velocity remains constant?
4. A spider runs back and forth in a straight line with the following data obtained:
t (s)
x (m)
0
4
2
4
6
8
10
12
6
12
12
8
4
0
A) Plot the position vs.
graph for the spider.
time
B) What do you think is happening during the
interval: t = 4s to t = 6 s? How do you know?
time
C) What do you think is happening during
time interval t = 6s to t = 12s? How do you
the
know?
D) Determine the spider’s average speed from t = 0s to t = 12s.
E) Determine the spider’s average velocity from t = 0s to t = 12s.
5. The spider is now seen to move in the following manner:
t (s)
x (m)
0
8
A) Plot the position vs. time graph for the
2
7
spider.
4
6
6
6
8
8
10
10
12
12
B) During what time interval is the spider traveling the fastest?
How do you know?
C) What do you think is happening during the time interval t = 4s to t = 6 s? How do you know?
D) Determine the spider’s average speed from t = 0s to t = 12s.
E) Determine the spider’s average velocity from t = 0s to t = 12s.
6. This data table shows information about two toy cars that were raced side-by-side.
t (s)
0
1
2
3
4
5
6
7
8
9
10
x1 (m)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
x2 (m)
0
1
2
3
4
5
6
7
8
9
10
A) Draw a motion
map of each one,
one above the
number line and
the other below the
number line
showing which car
is which.
B) Draw a graph of both cars on
C) Find the velocity of each toy car. Show your work.
the same graph.
Constant
VelocityVelocity
Model Worksheet 3:
Unit
I-A: Constant
Velocity vs.
Time Graphs
and Displacement
Worksheet
4
1. This motion map shows the position of an object once every second. From the motion map,
1. Use the motion
to answer the following questions.
answermap
the below
following:
a. Describe the motion of the object.
A) What can you conclude about the motion of the object?
B) Draw position-time and velocity-time graphs for the object on the graphs below.
Represent
Positionb.(m)
the motion with a quantitativeVelocity
x vs t graph.
(m/s)
c. Represent the motion with a quantitative v vs t graph.
Time (s)
Time (s)
d. Write a mathematical
expression
that represents
the relationship
C) Write mathematical
expressions
that represent
the relationships
betweenbetween
positionposition
and timeand time.
and between velocity and time for the object.
e. Write a mathematical expression that represents the relationship between velocity and time.
D) Describe what the area under the line in velocity-time graph represents. Crosshatch this
area.
f. Cross hatch the area under the velocity-time graph. What are the units of this area? Describe what
the area under the v-t graph represents and find its value.
2. Use the position-time data below to answer the following questions.
t (s) x (m)
0
0
1
2
2
4
Velocity
3
4(m/s)
4
7
5
10
6
10
7
10
8
5
9
0
A) Construct a position vs. time graph and a velocity vs. time graph for
this data.
Position (m)
Time (s)
14(s)
Time
C) Determine the displacement from t = 3.0 s to 5.0 s using the velocity-time graph. Show on
the graph what you did and explain your thinking.
D) Determine the displacement from t = 7.0 s to 9.0 s using the velocity-time graph. Show on
the graph what you did and explain your thinking.
E) Compare your results to the position-time graph. Do the two results match?
Unit I-A: Constant Velocity
Worksheet 5
1. Sketch the velocity vs. time graph and the motion map corresponding to the following
descriptions of the motion of an object.
6. Sketch velocity vs. time graphs and motion maps corresponding to the following descriptions of
the motion of an object.
Theobject
objectisismoving
movingaway
in thefrom
positive
direction
A)a.The
the origin
at a at
a constant
(steady)
speed.
constant
(steady)
speed.
Motion Map:
0
m
6. Sketch velocity vs. time graphs and motion+
maps corresponding to the following descriptions of
t
im
e
the motion of an object.
a. The object is moving in the positive direction at
a
(steady)
speed.still.
B) constant
The object
is standing
b. The object
Motion
Map: is standing still.
0
m
Motion Map:
0
m
+
tim
edescriptions of
6. Sketch velocity vs. time graphs and motion maps corresponding to the following
+
the motion of an object.
Theobject
objectmoves
is moving
in thethe
positive
direction
at speed,
C)a.The
towards
origin at
a steady
a
constant
(steady)
speed.
then it stands still.
t
im
e
b. The object is standing still.
Motion Map:
0
m Map:
Motion
c.
The object moves in the negative direction at a
0
m
steady speed for 10s, then stands still for 10s. +
tim
e
+
Motion Map:
0
m
tim
edescriptions of
6. Sketch velocity vs. time graphs and motion maps corresponding to the following
the motion of an object.
+
D) The object moves away from the origin at a steady speed,
a.
objectdirection
is moving
instill.
the positive
at origin at
b. The
standing
then
reverses
and
moves
back direction
towards the
a constant
(steady) speed.
the
same speed.
c. The object
Motion
Map: moves in the negative direction at a
0
mspeed
Motion
Map:for 10s, then stands still for 10s.
steady
0
m
d. The object moves in the positive direction at a
+
Motion Map:
steady
+
0
mspeed for 10s, reverses direction and moves
back toward the negative direction at the same
speed.
+
Motion Map:
0
m
Theobject
objectmoves
is standing
c.b.The
in thestill.
negative direction at+
a
steady speed for 10s, then stands still for 10s.
Motion
Map: moves in the positive direction at a
d. The object
0
mspeed
steady
Motion
Map:for 10s, reverses direction and moves
0
m
back toward the negative direction at the same
+
speed.
+
©Modeling
Instruction
Program
2009
5
Motion Map:
0
m
t
im
e
tim
e
tim
e
t
im
e
Constant Velocity ws 1
tim
e
Draw
the velocity
vs time
graphs
for for
an an
object
whose
motion
produced
time
graphs
2. Draw
the velocity
vs. time
graph
object
whose
motion
producedthe
theposition
distancevsvs.
time
shown shown
below below
at left.at left.
graphs
5.
6.
7.
8. For many graphs, both the slope of the line and the area between the line and the horizontal axis
have
meanings.
3.
For physical
many graphs,
both the slope of the line and the area between the line and the horizontal
a. What
doesphysical
the slope
of a position time graph tell you about the motion of an object?
axis have
meanings.
A) What does the slope of a position time graph tell you about the motion of an object?
b. Looking at the velocity time graphs, determine the units for a square of area on the graph.
B) Looking at the velocity time graphs, determine the units for a square of area on the graph.
c. What
does
the the
areaarea
under
the the
velocity-time
graph
tell
C) What
does
under
velocity-time
graph
tellyou
youabout
aboutthe
themotion
motion of
of an
an object?
object?
©Modeling Instruction Program 2009
5
Constant Velocity ws2
x
t
Unit I-A: Constant
Velocity
Worksheet 6
v
Motion map:
Fill in the blank boxes with the correct complete information.
Constant Velocity Particle Model Worksheet
4:
t
Multiple Representations of Motion
x - t Graph
v - t Graph
Motion Map
Given one motion representation, supply the missing motion representations.
1.
Written description:
2.
x
Written description:
x
3.
t
Written description:
x
v
t
Motion map:
t
v
Motion map:
3
6
8
t (s)
t
v
Motion map:
2.
Written description:
t
x
3.
Written description:
x
4.
Written description:
t
x
v
3
6
8
t (s)
Motion map:
t
v
Motion map:
t
t
+2
v
(m/s)
-1
Motion map:
2
4
4.
Written description:
x
t
+2
v
Motion map:
8 t (s)
Written Description
The object starts at 2m
and moves with a
constant positive velocity
for two seconds, then
stops for 2s, then returns
past the starting point at
a faster speed in 2s.
x - t Graph
v - t Graph
Motion Map
Written Description
The object moves with a
constant positive velocity
for 4 seconds. Then, it
stops for 2 seconds and
returns to the initial
position in 2 seconds.
Object A starts 10m to
the right of zero and
moves to the left at 2m/s.
Object B starts at zero
and moves to the right at
3 m/s
Unit I-A: Constant Velocity
Review Worksheet
Constant V
Vocabulary Terms
Coordinate System
Displacement
Distance
Scalar Quantity
Vector Quantity
Motion Map
Average Velocity
Average Speed
Instantaneous Velocity
Re
1. Consider the following position vs. tim
Review Problems
a. De
1. Consider the position vs. time graph shown.
A) Determine the average velocity of the object.
b. Wr
ob
B) Write the mathematical equation to describe the motion of the
object.
C) Using the equation for the graph, find the position of the object at
t = 10 s.
2. Shown below is a velocity vs. time gra
4. A basketball initially travels at 3 m
2. Consider the following v vs. t graph.
A) Describe the behavior of the object.
a. De
B) Draw a motion map to model the behavior of the object.
C) How far did the object travel in the interval t = 1s to t = 2s?
b. Draw a corresponding position vs. time
from the
origin.a quantitative motion map
b. Draw
0
m
c.
D) What is the TOTAL displacement? Explain how you got your answer.
c. How far did the ball travel from t
d. F
5. A racecar reaches a speed of 95 m
3. Use the following graph that represents the motion
of a runner for the next 3 questions.
A) Describe the motion of the runner during the 25second time interval.
B) What distance does the runner travel from 0 – 25
seconds?
C) What is the runner’s displacement from 0 – 25 seconds?
D) What is the runner’s average speed from 0 – 25 seconds?
E) What is the runner’s average velocity from 0 – 25 seconds?
4. A racecar travels at a speed of 95 m/s. How much time does it take to reach the finish line
500 m away?
5. A hummingbird averages a speed of about 28 miles/hour (Cool facts: They visit up to 1000
flowers per day, and reach maximum speed while diving ... up to 100 miles/hour!). Rubythroated hummingbirds take a 2000-mile journey when they migrate, including a non-stop trip
across Gulf of Mexico in which they fly for 18 hours straight! How far is the trip across the Gulf
of Mexico?
6. Dr. Foster and Mr. Padilla are out skateboarding. The table shows their motion over time.
Use the table to answer the questions.
Time
(s)
Position
Dr. Foster
(m)
Position
Mr. Padilla
(m)
0
0
45
5
10
40
10
20
35
15
30
30
20
40 Position (m) 25
25
50
20
30
60
15
A) Draw a motion map for each skateboarder (one above,
one below the line).
B) Draw a position-time graph for each skateboarder.
C) Does either one of the
move at a constant velocity? How
administrators
do you know?
D) Do Dr. Foster and Mr. Padilla
same direction? How do you
move in the
know?
E) What is happening at time = 15s?
F) How fast is Dr. Foster going? How fast is Mr. Padilla going? Show your work.
G) Do you want to join in the fun and skateboard with your administrators?
Time (s)