Name: ____________________________________________________
Physics Unit I: Motion
Subunit B: Constant Acceleration
Equations
Variables, Units
NOTES:
Period: ______
Unit I-B Objectives
What you should know when all is said and done
1. Given a 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 the corresponding a vs. t graph
d. draw a motion map for the object (including v and a vectors)
e. determine the instantaneous velocity of the object at a given time
2. Given a v vs. t graph, you should be able to:
a. describe the motion of the object (direction of motion, acceleration)
b. draw. the corresponding x vs. t graph
c. draw the corresponding a vs. t graph
d. draw a motion map for the object (including v and a vectors)
e. write a mathematical model to describe the motion
f. determine the acceleration
g. determine the displacement for a given time interval
3. You should be able to determine the instantaneous velocity of an object in three ways:
a. determining the slope of the tangent to an x vs. t graph at a given point.
b. using the mathematical model v = at + v0
c. using the mathematical model v2 = v02 + 2ax
4. You should be able to determine the displacement of an object in three ways:
a. finding the area under a v vs. t curve
b. using the mathematical model x = ½ at2 + v0t
c. using the mathematical model v2 = v02 + 2ax
5. You should be able to determine the acceleration of an object in five ways:
a. finding the slope of a v vs. t graph
b. using the mathematical model a = v/t
c. rearranging the mathematical model x = ½ at2 + v0t
d. rearranging the mathematical model v = at + v0
e. rearranging the mathematical model v2 = v02 + 2ax
Unit 1B Reading/Notes
In the opening lab, the motion of an object rolling down an inclined ramp was
investigated. We did not concern ourselves with the rotational motion of the object and
instead imagined it as a point particle. By recording the position of the particle at equal
time intervals, a position vs. time graph was generated. The shape of the graph
appeared to be a top opening parabola.
In the constant velocity model, we established that the slope of the position-time graph
(change in position divided by change in time) is the average velocity during the time interval. The
slope of a straight line can be taken at any two points on the line because the slope is constant. So
what does the fact that the slope is not constant in our new experiment mean?
Slope of the curved graph tells me that:
_______________________________________________________________________________
The average velocity can also be found for a curved position-time graph by the same method as for a
linear graph, by taking the total change in position over a time interval. However, on a curved graph,
the slope is constantly changing at each point, so the average velocity often isn’t a very good
description of the object’s motion. In order to find the velocity of the object at any given instant, or the
instantaneous velocity, a line can be drawn through two points to first find the average velocity. The
slope of this line will be approximately equal to the instantaneous velocity at some point exactly
halfway in between these two points. (An average is also like taking the two instantaneous velocities,
adding them together and dividing by two: vave = (vi + vf)/2.)
The closer the two points are on the curve, the closer the average velocity will be to
the instantaneous velocity. As the two points get closer and closer, they reach a
point where they meet and the line going between them is now tangent to the line at
that point. A tangent line touches a curve at just one point without crossing from one
side of the curve to the other. The slope of the tangent tells us the velocity at the time
corresponding to the point on the graph to which the tangent is drawn. Since the
velocity is always changing, what we find is the instantaneous velocity.
Your math teacher may have taught you how to find the equation for a parabola, but here in physics,
we are going to use a concept called “linearization” to simplify the analysis of a graph by recognizing a
pattern and creating a test plot that represents what model we expect the data to follow. You may
know that a parabola is a graph that can also be called a “quadratic function”, or is represented by an
equation of y vs. x2. If we make a test-plot of position vs. time squared using our lab data, we will find
that the graph becomes linear; this means we can write the equation of the new test plot in the form
y = mx + b using position as the y variable and time squared as the x. Therefore we can state that the
relationship between the variables was that position is directly proportional to time squared. Let’s try it!
Finding instantaneous velocity by determining the slopes of tangents will allow us to
create a plot of velocity vs. time. Making a graph of instantaneous velocity vs. time
yields a linear graph.
A linear graph of velocity vs. time (like the one at right) tells me that:
_______________________________________________________________________________
The slope of the velocity vs. time graph is the change in velocity divided by change in time. It tells us
how much the velocity changes during each time interval. A large slope means that the velocity
changes a lot every second, whereas a small slope means that the velocity changes a small amount
each second. Since the rate of change in velocity is a useful idea, we give it a name: acceleration.
Because both speeding up AND slowing down represent changes in velocity, they are both called
acceleration (no need for the odd term “deceleration”).
Acceleration:
_______________________________________________________________________________
Plotting the instantaneous velocity by determining the slopes of tangents will always work, but it is
often a bit tedious (ok, a lot tedious!). So remember: When the average velocity is determined for a
time interval, we find that this average velocity is identical to the instantaneous velocity at the time in
the middle of the interval. For example, if the average velocity from t = 2s to t = 4s is 10 m/s, the
instantaneous velocity is 10 m/s at the time in the middle of the interval, t = 3s. Let’s test this
technique out on worksheet 1 to help us determine what the slope from our new linearized graph
represents!
Unit I-B: Constant Acceleration
Worksheet 1
t
(s)
0.0
x
(cm)
0.0
1.0
5.0
2.0
20.0
3.0
45.0
4.0
80.0
5.0
125.0
6.0
180.0
t2
(s2)
t
(s)
x
(cm)
vave
(m/s)
tmp
(s)
The data to the left are for a marble rolling
from rest down an incline. Use the
position-time data given in the data table to
do the following:
A) Plot a position vs. time graph for the
data on the axes below, using the entire
graph area. Label the graph clearly.
B) Complete the rest of the data table. For
the four columns on the right, you are
calculating the change from the previous
row to the subsequent row. tmp means the
mid-point time of the interval.
Position (cm)
0
1
2
3
4
5
6
7
8
Time (s)
C) Graph and label a test plot (to
linearize the data) on the grid, then
write the equation of the line
below.
D) Plot and label a velocity vs.
time graph on the second grid. Be
sure to plot the time column that
makes your v vs. t graph an
instantaneous velocity vs. time
graph. Write the equation of the
line below.
g. Answer the 11 questions on
the following page!
1. What is the meaning of the slope of a position vs. time graph?
2. What is happening to the slope of your position vs. time graph as time goes on?
3. Explain what your answers to questions 1 and 2 tell you about the motion of the marble.
4. What is the meaning of the slope of your velocity vs. time graph? Explain!
5. Compare the slope of your velocity vs. time graph to the slope of your position vs. time2 graph.
What does this tell you about the slope of your position vs. time2 graph?
6. Write an equation that relates velocity and time for the ball using the mathematical analysis of your
velocity vs. time graph.
7. Write an equation that relates position and time for the ball using the mathematical analysis of your
position vs. time2 graph.
8. On the position vs. time graph, draw a line which connects the data point at t = 0 to the data point
at t = 6 s and calculate the slope of this line. Explain what the slope of this line tells you about the
motion of the ball.
9. On the position vs. time graph, draw a line which connects the data point at t = 2 s to the data point
at t = 4 s. Calculate the slope of this line. Explain what the slope of this line tells you about the motion
of the ball.
10. On the position vs. time graph, draw a line tangent to the graph at t = 3 s. Calculate the slope of
this line. Explain what the slope of this line tells you about the motion of the ball.
11. Compare the slopes you have calculated in questions 8, 9, and 10. Explain the results of your
comparison.
Unit I-B: Constant Acceleration
Worksheet 2
1. Accelerating objects are objects that are changing their velocity. Name the three controls on an
automobile that cause it to accelerate.
2. An object must be accelerating if it is moving _____. Circle all that apply.
A) with changing speed
D) in a circle
B) extremely fast
E) downward
C) with constant velocity
F) none of these
3. If an object is NOT accelerating, then one knows for sure that it is
A) at rest
C) slowing down
B) moving with a constant speed
D) maintaining a constant velocity
4. An object with an acceleration of 10 m/s2 will ____. Circle all that apply.
A) move 10 meters in 1 second
C) move 100 meters in 10 seconds
B) change its velocity by 10 m/s in 1 s
D) have a velocity of 100 m/s after 10 s
5. Ima Speedin puts the pedal to the metal in her Porsche and accelerates from 0 to 60 mi/hr in 4
seconds. Her acceleration is
A) 60 mi/hr
C) 15 mi/hr/s
B) 15 m/s/s
D) -15 mi/hr/s
Motion
in One Dimension
6. A car speeds up from rest to +16 m/s in 4 s. Calculate the acceleration.
Acceleration as a Vector Quantity
Acceleration, like velocity, is a vector quantity. To fully describe the acceleration of an object, one must
describe
theslows
direction
the acceleration
A general
rule ofthe
thumb
is that if an object is moving in
7. A car
downoffrom
+32 m/s to vector.
+8 m/s in
4 s. Calculate
acceleration.
a straight line and slowing down, then the direction of the acceleration is opposite the direction the object
is moving. If the object is speeding up, the acceleration direction is the same as the direction of motion.
9.8. Read
Read the following
statements and
west,
north
or or
south)
of of
the
following statements
andindicate
indicatethe
thedirection
direction(up,
(up,down,
down,east,
east,
west,
north
south)
theacceleration
accelerationvector.
vector.
10.
a.
Description of Motion
A car is moving eastward along Lake Avenue and increasing its speed
from 25 mph to 45 mph.
b.
A northbound car skids to a stop to avoid a reckless driver.
c.
An Olympic diver slows down after splashing into the water.
d.
A southward-bound free quick delivered by the opposing team is
slowed down and stopped by the goalie.
e.
A downward falling parachutists pulls the chord and rapidly slows
down.
f.
A rightward-moving Hot Wheels car slows to a stop.
g.
A falling bungee-jumper slows down as she nears the concrete
sidewalk below.
The diagram at the right portrays a Hot Wheels track
designed for a phun physics lab. The car starts at
point A, descends the hill (continually speeding up
from A to B); after a short straight section of track, the
car rounds the curve and finishes its run at point C.
Dir'n of
Acceleration
a. rightward, rightward
c. leftward, rightward
e. rightward, zero
b. rightward, leftward
d. leftward,
leftward
Describing Motion
Graphically
f. leftward, zero
Study Lessons 3 and 4 of the 1-D Kinematics chapter at The Physics Classroom:
http://www.physicsclassroom.com/Class/1DKin/1KinTOC.html
Renatta Oyle's
car has an oil leak and leaves a trace of oil drops on the streets as she drives through
9.Glenview.
Renatta Oyle's
car
an oil leak
and
leaves
trace of oil
drops
on the
as the
she drives
A
study
of has
Glenview's
streets
reveals
theafollowing
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Match
the streets
trace 9-11)
with
MOP Connection:
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Graphing:
sublevels
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Motion
inAlamo
One Dimension
Name:
through
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reveals
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Match
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verbal descriptions given below. For each match, verify your reasoning.
verbal
descriptions
given
below.
For
each
match,
verify
your
reasoning.
1. The slope of the line on a position vs. time graph reveals information about an object's velocity. The
magnitude
(numerical
value) of the slope is equal to the object's speed and the direction of the slope
Diagram
A:
Describing
Motion
(upward/ + or downward/ -) is the same
as the direction
of theGraphically
velocity vector. Apply this
understanding to answer the following questions.
Study Lessons
3 and 4 of the 1-D Kinematics chapter at The Physics Classroom:
B: line
a. ADiagram
horizontal
means
.
5.
2.
http://www.physicsclassroom.com/Class/1DKin/1KinTOC.html
b. A straight diagonal line means
.
MOP
Connection:
Kinematic
Graphing:
sublevels
1-11
(emphasis
on sublevels 9-11)
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curved C:
line means
.
The
slope
of the line
on a position vs. time graph reveals information
about anDiagram
object's velocity. The
d.1. A gradually
sloped
line means
.
Verbal
Description
magnitude
(numerical
value)
of
the
slope
is
equal
to
the
object's
speed
and
the
direction of the slope
Description
Diagram
i. Verbal
Renatta
was
driving
with a slow constant speed, accelerated
to rest,
e. A steeply
sloped
line
means
.
+was
or driving
downward/a slow
-) is the same
as the
direction
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i.(upward/
Renattaat
speed,
decelerated
to of
rest,
remained
remained
rest for 30with
s, and thenconstant
drove very
slowly
at a constant
speed. at
understanding
to
answer
the
following
questions.
rest
for
30
s,
and
then
drove
very
slowly
at
a
constant
speed.
Reasoning:
The motion of several objects is depicted on the position vs. time graph. Answer the following
a. A
horizontal line means
.
questions. Reasoning:
Each question may have less than one, one, or more than one answer.
ii.
Renatta rapidly decelerated from a high speed to a rest position, and then
b. A straight diagonal line means
.
slowly
accelerated
a moderate
speed.
a. Which
object(s)tois(are)
at rest?
c.Reasoning:
A
curved
linedecelerated
means
.
ii.
from a high speed to a rest position, and then slowly
b. Renatta
Which rapidly
object(s)
is(are) accelerating?
accelerated
to
a
moderate
speed.
d.
gradually
sloped
line
means
iii.
Renatta
was
driving
at
a moving?
moderate speed and slowly accelerated. .
c. A
Which
object(s)
is(are)
not
Reasoning:
e.Reasoning:
steeply
slopedchange(s)
line means
d. A
Which
object(s)
its direction?
2.
.
e. Which object is traveling fastest?
The
motionwas
of several
objects
is depicted
on
the position
vs. time graph. Answer the following
Renatta
driving at
a moderate
speed and
slowly
accelerated.
10.iii.
The
motionmoving
of several
objects
is have
depicted
on
the
position
vs.ortime
graph.
the following
f. Which
object
ismay
traveling
slowest?
questions.
Each
question
less
than
one,
one,
more
thanAnswer
one answer.
questions. Each question may have less than one, one, or more than one answer.
g. Reasoning:
Which
is(are)
moving
in rest?
the same direction as object B?
a. object(s)
Which
object(s)
is(are)
_____
A)
Which
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at rest?at
3.
b.line
Which
object(s)
is(are)
B)the
Which
object(s)
is(are)
accelerating?
The _____
slope of
on a velocity
vs.
time accelerating?
graph reveals information about an object's acceleration.
Furthermore,
the
area
under
the
line
is
equal
to the object's displacement. Apply this understanding
_____ C) Which
object(s)
is(are)is(are)
not moving?
c. Which
object(s)
not moving?
to answer the following questions.
_____ D) Which
object(s)
change(s)
its direction?
d.line
Which
object(s)
change(s)
its direction? .
a. A horizontal
means
_____ E) Which
objectobject
is traveling
fastest?fastest?
e. Which
is traveling
b. A straight diagonal
line means
.
_____
F) Which
moving
object
traveling
slowest? slowest?
f. sloped
Which
moving
is traveling
c. A
gradually
line
meansisobject
.
_____
G) Which
is(are) moving in the same direction
as
d. A
steeply
slopedobject(s)
line means
.
object B? g. Which object(s) is(are) moving in the same direction as object B?
4.
© The Physics Classroom, 2009
Page 10
3. motion
The slope
of theobjects
line on
velocityby
vs.a time
graph
reveals
information
about
an
object's acceleration.
The
of several
is adepicted
velocity
vs. time
graph.
Answer the
following
The
motion
of
several
objects
is
depicted
by
a
velocity
vs.
time
graph.
Answer
the
following
Furthermore,
the area
the than
line is
equal
object's
displacement.
Apply questions.
this understanding
questions.
Each question
mayunder
have less
one,
one,toorthe
more
than one
answer.
Each
question
may
have
less
than
one,
one,
or
more
than
one
to answer the following questions.
a. Which object(s) is(are) at rest?
answer.
a.
line is(are)
meansat rest?
.
_____ A
A) horizontal
Which object(s)
b. Which object(s) is(are) accelerating?
b. A
diagonal
line accelerating?
means
.
_____
B) straight
Which object(s)
is(are)
Which object(s)
is(are)
not
moving?
c.c. A
sloped
line
means
.
_____
C) gradually
Which object(s)
is(are)
not
moving?
Which object(s)
its direction?
d.d. A
slopedchange(s)
line
means
.
_____
D) steeply
Which object(s)
change(s)
its direction?
e. Which accelerating object has the smallest acceleration?
_____ E) Which accelerating object has the smallest acceleration?
4. The
of several
objects
is depicted
by a velocity vs. time graph. Answer the following
f. motion
Which object
has the
greatest
acceleration?
_____
F) Which
object
has themay
greatest
acceleration?
questions.
Each
question
have
less than one, one, or more than one answer.
g. G)
Which
object(s)
is(are)
moving
_____
Which
object(s)
is(are)
movingininthe
thesame
samedirection
direction as
as object
object E?
E?
a. Which object(s) is(are) at rest?
b. Which object(s) is(are) accelerating?
c. Which object(s) is(are) not moving?
d. Which object(s) change(s) its direction?
Graphing Summary
s/1DKin/1KinTOC.html
Motion in One Dimension
Constant Velocity
e(emphasis
1-D Kinematics
chapter
at The Physics Classroom:
Object moves in + Direction
on sublevels
9-11)
www.physicsclassroom.com/Class/1DKin/1KinTOC.html
Velocity Dir'n:
+
or -
Constant Velocity
Object moves in - Direction
Constant +
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Velocity Dir'
ty
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nematic
sublevels
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in + Direction
Speeding up or
Study Lessons 3 and 4 of the 1-D Kinematics chapter at The Physics Classroom:
Unit I-B: Constant Acceleration
http://www.physicsclassroom.com/Class/1DKin/1KinTOC.html
ConstantWorksheet
+ Acceleration
3
or tion
Velocity
Dir'n:
+ or Constant
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Object
moves
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e 1-D Kinematics chapter at The Physics Classroom:Constant + Acceleration
ation
Constant - Acceleration
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www.physicsclassroom.com/Class/1DKin/1KinTOC.html
ection
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nematic Graphing: sublevels 1-11 (emphasis on sublevels
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tion
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2009
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moves
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x
Page
20
C) Moving in the - direction and speeding up
Velocity Dir'n: + or Velocity Dir'n: + or Speeding up or Slowing Down?
Speeding up or Slowing Down?
© The Physics Classroom, 2009
Constant - Acceleration
Constant - Acceleration
Object moves in - Direction
Object moves in + Direction
x
Page 20and slowing down
D) Moving in the - direction
Velocity Dir'n: + or Velocity Dir'n: + or Speeding up or Slowing Down?
Page 20
Speeding up or Slowing Down?
Page2009
20
© The Physics Classroom,
Constant - Acceleration
Object moves in - Direction
Velocity Dir'n:
+
or -
Speeding up or Slowing Down?
down
Acceleration
-
Constant - Acceleration
Object moves in - Direction
Velocity Dir'n:
+
or -
Speeding up or Slowing Down?
Acceleration
tion
Acceleration
MOP Connection:
Kinematic Graphing: sublevels 1-11 (emphasis on sublevels 9-11)
Motion
maps show the position of Velocity
the object
at equal time intervals. For uniformly accelerated motion,
Physics
Classroom:
Velocity Dir'n: + or Dir'n: + or the spacing
between dots increases or decreases depending on whether the object is speeding up or
Graphing
Summary
s/1DKin/1KinTOC.html
Speeding
up or
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(velocity
vectors)
drawn
on the dots are
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in the direction of Constant +
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vectors
can also
be moves
drawn
next
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e(emphasis
1-D Kinematics
chapter
at The Physics
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Object
in
+ Direction
Object
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in - Direction
Object
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+ Acceleration
Constant
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the
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ation
Constant -isAcceleration
moves in - Direction
Object moves in - Direction
Object moves
www.physicsclassroom.com/Class/1DKin/1KinTOC.html
ection
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Object moves in + Direction
on
ion
-
Down?
2. While
cruising
along a
darkxstretch of highway at 25 m/s (55 mph), you see, at the fringes of your headlights, that a bridge
Velocity Dir'n: + or Velocity Dir'n: + or has been washed out. You apply the brakes and come to a stop in 4.0s. Assume the clock starts the
Speeding
up
or
Slowing
Down?
Speeding
up or Slowing Down?
instant you hit the brakes.
Constant - Acceleration
Object moves in - Direction
Constant - Acceleration
Object moves in + Direction
A) Construct a motion map that represents the motion described above, including position, velocity,
and acceleration. Clearly demonstrate how you can determine the direction (sign) of the
acceleration from the motion map representation.
Page 20
x
B) Construct a qualitative x vs. t graph of the situation described, and a quantitatively accurate
v vs. t graph to describe the situation.
© The Physics Classroom, 2009
Position (m)
Velocity (m/s)
2009
Page 20
Time (s)
Time (s)
C) On the v vs. t graph at right, graphically represent the car’s displacement during braking.
D) Utilizing the graphical representation, determine how far the car traveled during braking. (Show
work!)
Acceleration (m/s2)
E) Determine the car’s acceleration, then sketch a
quantitatively accurate a vs. t graph.
F) Using the equation you developed for displacement of an
accelerating object determine how far the car traveled during
0
Time (s)
braking. (Show your work.)
G) Compare your answers to D and F.
v (m/s)
12
3. Use the following graph to answer questions for object C.
A) Give a written description of the motion.
9
6
B) Sketch a motion map. Be sure to include both velocity and
acceleration vectors.
3
0
2
4
6
8 t (s)
2
4
6
8
-3
-6
xC) Determine the displacement from t = 0s to t = 4 s.
-9
-12
D) Determine the displacement from t = 4 s to t = 8 s.
E) Determine the average acceleration of the object’s motion.
v (m/s)
4. Use the following graph to answer questions for object D.
A) Give a written description of the motion.
8
6
4
2
B) Determine the displacement from t = 0s to t = 2 s.
0
-2
-4
C) Determine the displacement from t = 2 s to t = 4 s.
-6
-8
D) Determine the total displacement.
E) Determine the object’s acceleration at t = 2 s.
t (s)
25
0
Velocity (m/s)
5. A car accelerates from rest to a speed of 20 m/s in a time of
5.0 seconds.
A) Sketch a velocity-time graph showing the motion of the car.
B) What is the acceleration of the car?
C) What distance will it travel as it accelerates? How do you
know?
14
Time (s)
Unit I-B: Constant Acceleration
Worksheet 4
1. The graph at right represents the motion of
Object A.
A) Where on the graph above is the object
moving most slowly? How do you know?
B) Between which points is the object speeding
up? How do you know?
C) Between which points is the object slowing down? How do you know?
D) Where on the graph above is the object changing direction? How do you know?
2. A) For each line segment shown on the graph, identify if the acceleration and velocity are positive,
negative, or zero. Then decide if the object would be speeding up, slowing down, not moving, or
moving with constant velocity. Fill in the chart to show your answers.
Velocity (m/s)
v
Description
A
B
A
a
B
C
C
F
Time (s)
G
D
E
D
E
F
G
B) On the graph there is a dark black dot between sections C and D. What is the velocity at that exact
moment in time? What is the object doing? Is it still accelerating? Why or why not?
Motion in One Dimension
Name:
3. The motion of a two-stage rocket is portrayed by the following velocity-time graph. Several students
Interpreting Velocity-Time Graphs
analyze the graph and make the following statements. Indicate whether the statements are correct or
The motion
of a
two-stage
rocket
is portrayed
by thefeatures
following
velocity-time
incorrect.
Justify
your
answers
by referring
to specific
about
the graph.graph.
theisgraph
and
the following
statements.
Indicate
whether
statements
A)Several
After 4 students
seconds,analyze
the rocket
moving
inmake
the negative
direction
(i.e., down).
Correct?
Yes the
or No
are correct or incorrect. Justify your answers by referring to specific features about the graph.
Justification:
Student Statement
Correct?
Yes or No
1. After 4 seconds, the rocket is moving in the negative direction (i.e.,
down).
B) The rocket is traveling with a greater speed during the time interval from 0 to 1 second than the
Justification:
time interval
from 1 to 4 seconds. Correct? Yes or No
Justification:
2. The rocket is traveling with a greater speed during the time interval from
0 to 1 second than the time interval from 1 to 4 seconds.
C) The firstJustification:
engine burns out quickly, but provides a higher acceleration. Correct? Yes or No
Justification:
3. The rocket changes its direction after the fourth second.
Justification:
D) During the time interval from 4 to 9 seconds, the rocket is moving in the positive direction (up) and
slowing down. Correct? Yes or No
Justification:
4. During the time interval from 4 to 9 seconds, the rocket is moving in the
positive direction (up) and slowing down.
Justification:
E) At nine seconds, the rocket has returned to its initial starting position. Correct? Yes or No
Justification:
5. At nine seconds, the rocket has returned to its initial starting position.
Justification:
F) During the time interval from 9 to 14 seconds, the rocket is slowing down in the negative direction.
Correct? Yes or No
Justification:
© The Physics Classroom, 2009
Page 19
4. The following graph shows Shopping Sandy’s velocity as she races up and down the walkway in
North Star Mall trying to find the perfect pair of capris. Answer the questions below.
Velocity (m/s)
6
4
2
Time (s)
0
-2
-4
-6
0
4
8
12
16
20
24
28
32
36
40
A) Describe Sandy’s motion.
B) During what time periods is Sandy accelerating? Find Sandy’s acceleration for each time period.
C) What is Sandy’s displacement from t = 6s to t = 14s?
D) What is Sandy’s displacement from t = 14 s to t = 22 s?
E) What is Sandy’s displacement from t = 30s to t = 36s?
Unit I-B: Constant Acceleration
Worksheet 5
1. A bicycle starts from rest and reaches a speed of 2.5 m/s during
a time of 5 seconds.
A) Draw a velocity-time graph for the bicycle.
5
Velocity (m/s)
B) What was the bicycle’s acceleration?
C) How far did the bicycle travel during this time?
14
2. A resting cat gets startled by a dog. It turns and runs with an
acceleration of 8 m/s2, reaching full speed in only 0.8 seconds.
A) What is its final velocity?
5
Time (s)
Velocity (m/s)
B) What distance does the cat cover as it accelerates?
14
C) Make a velocity-time graph for the cat.
3. An old clunker car can accelerate from rest to a speed of
28 m/s in 20 s.
A) Draw a velocity-time graph for the car.
5
Time (s)
Velocity (m/s)
B) What is the average acceleration of the car?
C) What distance does it travel in this time?
14
Time (s)
4. A motorcycle goes from 15 m/s to a dead stop in 3 seconds.
A) Draw a velocity-time graph for the motorcycle.
5
Velocity (m/s)
B) What is its acceleration?
C) What distance will it travel?
14
5. An ostrich has an acceleration of -2 m/s2. If it is initially traveling
at a velocity of +7.5 m/s,
A) How long will it take to completely stop?
5
Time (s)
Velocity (m/s)
B) What distance will it travel?
14
Time (s)
C) Draw a velocity-time graph for the ostrich.
6. A Hot Wheels car accelerates down a 5-meter long ramp. If the car takes 2.5 seconds to reach the
bottom of the ramp,
Velocity (m/s)
A) What is its acceleration?
5
B) What is the speed of the car at the bottom of the ramp?
C) Draw a velocity-time graph for the car.
14
Time (s)
5
7. An airplane accelerates down a runway at 3.0 m/s2 for 33 s until
is finally lifts off the ground.
A) Determine the distance traveled before takeoff.
Velocity (m/s)
B) What is the airplane’s lift-off velocity?
14
Time (s)
C) Draw a velocity-time graph for the airplane.
8. A bullet leaves a rifle with a muzzle velocity of 521 m/s.
While accelerating through the barrel of the rifle, the bullet
moves a distance of 0.840 m.
A) Determine the acceleration of the bullet.
5
Velocity (m/s)
B) How long was the bullet traveling inside the barrel of the gun?
14
C) Draw a velocity-time graph for the bullet.
Time (s)
Unit I-B: Constant Acceleration
Worksheet 6
1. A bear spies some honey and takes off from rest, accelerating at a rate of 2.0 m/s2. If the honey is
16 m away, how fast will he be going when he reaches it?
Equation:
v0 =
Δx =
v=
a=
t=
2. At t = 0 s, a car has a speed of 30 m/s. At t = 6 s, its speed is 14 m/s. What is its average
acceleration during this time interval?
Equation:
v0 =
Δx =
v=
a=
t=
3. A bus initially moving at 20 m/s slows at a rate of 4 m/s each second.
A) How long does it take the bus to stop?
Equation:
v0 =
Δx =
v=
a=
t=
B) How far does it travel while braking?
Equation:
4. A physics student skis down a hill, accelerating at a constant 2.0 m/s2. If it takes her 15 s to reach
the bottom, what is the length of the slope?
Equation:
v0 =
Δx =
v=
a=
t=
5. As a car passes, a dog runs down his driveway to chase it with an initial speed of 5 m/s, and
uniformly increases his speed to 10 m/s in 2 s.
A) What was his acceleration?
Equation:
v0 =
Δx =
v=
a=
t=
B) How long is the driveway (TOTAL displacement)?
Equation:
6. A mountain goat starts a rockslide and the rocks crash down the slope 100 m. If the rocks reach the
bottom in 5 s, what is their acceleration?
Equation:
v0 =
Δx =
v=
a=
t=
7. A car whose initial speed is 30 m/s slows uniformly to 10 m/s in 5 seconds.
A) Determine the acceleration of the car.
Equation:
v0 =
Δx =
v=
a=
t=
B) Determine the displacement of the car.
Equation:
Unit I-B: Constant Acceleration
Worksheet 7: Wile E. Coyote on the
Planet Newtonia
Wile E slipped off the edge of a tall building and was
photographed at one-second intervals as he underwent free
fall. Complete the table below, plot final velocity vs. time,
then answer the questions.
T
y
Vave
Vf
(s)
(m)
(m/s)
(m/s)
0
1
2
3
4
5
1. Write the equation for the graph above.
2. Using your graph, determine the value of the acceleration.
3. Using your graph, determine the displacement during the
first 3 s.
4. Using the mathematical model, determine the
displacement during the first 3 s.
Wile E. Coyote in Free Fall on Newtonia’s Moon
Wile E. slips off the edge of a cliff and was
photographed at one-second intervals as he
underwent free fall. Complete the table below, plot
final velocity vs. time, then answer the questions.
T
y
Vave
Vf
(s)
(m)
(m/s)
(m/s)
0
1
2
3
4
5
6
1. Write the equation for the graph above.
2. Using your graph, determine the value of the
acceleration.
3. Using your graph, determine the displacement
during the first 3 s.
4. Using the mathematical model, determine the
displacement during the first 3 s.
5. How does the gravity on the moon compare to that of Newtonia?
Unit I-B: Constant Acceleration
Worksheet 8: Free Fall
1. A ball is dropped from the top of the Leaning Tower of Pisa, 70 m above the ground.
A) How long does it take to hit the ground?
Equation:
v0 =
Δx =
v=
a=
t=
B) What will be its velocity the moment it hits the ground?
Equation:
2. Now the ball is thrown down from the tower (still 70 m high) with an initial velocity of 3 m/s.
A) How fast will it be going when it reaches the ground?
Equation:
v0 =
Δx =
v=
a=
t=
B) How long will it take to reach the ground?
Equation:
3. A ball is thrown straight up into the air with an initial velocity of 15 m/s.
A) How high does it go?
Equation:
v0 =
Δx =
v=
a=
t=
B) What is the ball’s hang-time (how long is it in the air)?
Equation:
4. A kangaroo jumps to a vertical height of 2.5 m. What was its hang-time?
Equation:
v0 =
Δx =
v=
a=
t=
5. You drop a rock in a well and see it hit the bottom in 2 seconds. How deep is the well?
Equation:
v0 =
Δx =
v=
a=
t=
6. A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the
moon is 1.67 m/s2. Determine the time for the feather to fall to the surface of the moon.
Equation:
v0 =
Δx =
v=
a=
t=
7. Luke Autbeloe drops a pile of roof shingles from the top of a roof located 8.52 meters above the
ground. Determine the time required for the shingles to reach the ground.
Equation:
v0 =
Δx =
v=
a=
t=
8. Rex Things throws his mother's crystal vase vertically upwards with an initial velocity of 26.2 m/s.
Determine the height to which the vase will rise above its initial height.
Equation:
v0 =
Δx =
v=
a=
t=
9. Upton Chuck is riding the Giant Drop at Great America. If Upton free falls for 2.6 seconds, what will
be his final velocity and how far will he fall?
Equation:
v0 =
Δx =
v=
a=
t=
Unit I-B: Constant Acceleration
Worksheet 9: Stacks of kinematics curves
Given the following position vs. time graphs, sketch the corresponding velocity vs. time and
acceleration vs. time graphs.
For the following velocity vs. time graphs, draw the corresponding position vs. time and
acceleration vs. time graphs.
Unit I-B: Constant Acceleration
Review Worksheet
1. A) Give a written description to describe the motion of
this object.
B) Draw the motion map for the object. Include velocity
and acceleration vectors.
x
C) Describe how could you find the instantaneous velocity of the object at t = 2 s.
D) Assume the initial velocity was 8 m/s; determine the acceleration of the object.
Equation:
v0 =
Δx =
v=
a=
t=
E) Sketch a corresponding velocity-time graph.
2. Use the graph to answer the following questions.
A) Describe the motion of the object.
B) Determine the acceleration of the object from the graph.
C) Calculate the object's displacement from 0 to 6 seconds.
3. A car, initially at rest, accelerates at a constant rate of 4.0 m/s2 for 6 s. How fast will the car be
traveling at t = 6 s? (Show your work!)
Equation:
v0 =
Δx =
v=
a=
t=
4. A tailback initially running at a velocity of 5.0 m/s becomes very tired and slows down at a
uniform rate of 0.25 m/s2. How fast will he be running after going an additional 10 meters?
Equation:
v0 =
Δx =
v=
a=
t=
5. A cliff diver jumps from a cliff and lands in the ocean water after 3.5 seconds.
A) What is the height of the cliff?
Equation:
v0 =
Δx =
v=
a=
t=
B) What was the velocity of the diver upon entering the water?
Equation:
6. A ball player catches a ball 4 seconds after throwing it vertically upward.
A) With what speed did he throw it?
Equation:
v0 =
Δx =
v=
a=
t=
B) How high did it go?
Equation:
7. Using the graph, compare the following quantities for objects A and B. Is A > B, A < B, or A = B?
A) Displacement from 0 to 3 s ___________ How do you know?
B) Average velocity from 0 to 3 s ___________ How do you know?
C) Instantaneous velocity at 3 s ___________ How do you know?
8. For each of the position vs. time graphs shown below, draw the corresponding v vs. t, a vs. t , and
motion map.
x
x
x