Created by Luke Tunstall
Acceleration – 1-D Motion for Calculus Students (90 Minutes)
Learning Goals: Using graphs and functions, the student will explore the various types of
acceleration, as well as how acceleration relates to position, velocity, and speed.
Prior Knowledge: Students should be familiar with the concept of 1-D motion, graphs of time
versus position, and graphs of time versus velocity and speed. Students should also be able to
take derivatives, find critical points, and use concavity to describe functions.
NCDPI/AP Objectives:
Competency Goal 2: The learner will use derivatives to solve problems.
2.03: Interpret the derivative as a function: translate between verbal and algebraic
descriptions of equations involving derivatives.
2.05: Interpret the second derivative: identify the corresponding characteristics of the
graphs of ƒ, ƒ', and ƒ".
2.06: Apply the derivative in graphing and modeling contexts: interpret the derivative as
a rate of change in varied applied contexts, including velocity, speed, and acceleration.
Materials Needed:
 One large toy-car or truck (big enough for the class to see)
 Masking tape
 Graphing calculators for each student (or at least one for each group)
 Worksheet copies for each student
 Whiteboard/markers
Think-Pair-Share Warm-up: Write the following example on the board or overhead for
students to begin as they walk in.
Question: Find the inflection points and intervals of positive and negative concavity of the
position function, d(t) =
. Use a graph to support your answer. Simply looking at the
position graph, what do you think might be the significance of d’’(t)? Why?
Answer: The position function d(t) has an inflection point at t = 6. At that instance, d(t) changes
from concave down to concave up. This means that d’’(t) changes from negative to positive at t =
6. The significance of d’’(t) is that it gives us the acceleration of an object.
The students should spend 5 minutes working on the problem alone. Then, give them a few
minutes to discuss their answers with a designated partner. Finally, a pair should be chosen to
present or read aloud their answers. The ‘share’ aspect should include some discussion of d’’(t),
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and it’s likely that one of the students will guess that d’’(t) represents acceleration. This is the
ideal bridge into the formal lesson.
Instructional Activities:
Lecture: Begin the lesson with a simple lecture on basic concepts that students will use for their
group activities. The students will fill in the notes on their worksheet. Before writing out the
‘punch-line’ of each statement, ask for a student to volunteer an answer for what to write.
Acceleration is the rate at which velocity is changing. Thus, a(t) = v’(t) = d’’(t).
If velocity is measured in m/s, then acceleration would be in m/ .
The critical points of v(t) indicate possible changes in the sign of acceleration.
Before moving on to a demonstration, write the following example on the board for students to
complete as independent practice. Give the students a few minutes by themselves, and then ask
for a volunteer to come to the board and explain their answer.
Question: Find where the acceleration is changing if the velocity of an object is given by: v(t) =
– 4t – 2.
Answer: When t = (2/3), acceleration changes from negative to positive.
Demonstration: Before beginning the group activities, perform a simple demonstration in front
of the entire class. Inform the students that acceleration isn’t as intuitive as they might think. For
example, what does it mean for an object to have negative velocity and positive acceleration? Is
the object actually speeding up? You’ll do a demonstration of 5 various types of acceleration,
and the students will model each of these soon.
Place a straight piece of masking tape on the floor or on a table. Tell the students that this will be
our line of motion. Ask for a student or two come up to help stop the cars.
o
o
o
o
o
To the best of your ability, roll the car at a constant speed in any direction; a(t) = 0
Roll the car in the right direction so that it’s speeding up; +v(t), +a(t)
Roll the car in the right direction so that it slows down immediately; +v(t), –a(t)
Roll the car in the left direction so that it’s speeding up; –v(t), –a(t)
Roll the car in the left direction so that it’s slowing down immediately; –v(t), +a(t)
Group Activities, Part 1: Tell the students it’s time to form groups so they can consider these
types of acceleration more seriously. Following the lesson-plan, I’ve included a page of slips to
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be copied (according to the number of students) and then cut out. Hand one slip to each student,
and then tell the students to find their fellow group members. The functions are listed below:
Group 1: d(t) =
v(t) = 6t
a(t) = 6
Group 2: d(t) =
v(t) = -8t
a(t) = -8
Group 3: d(t) =
v(t) =
a(t) =
Group 4: d(t) =
v(t) =
a(t) =
Group 5: d(t) = 8t
v(t) = 8
a(t) = 0
The students should work in groups on Part 1 of their worksheet for about 15 minutes, while you
circulate and answer questions.
Group Activities, Part 2 Jigsaw: After the groups have finished, scramble the groups so that
each new group has exactly one member from each possible function.
Now, for the next 20 minutes, the students should go over their answers as a team and fill out the
rest of the worksheet, except for the last practice section. Each member should have a turn
explaining their previous group’s work.
Whole-Class Discussion: After the groups have finished for the second time, the class should
come back together for a discussion and to redo the demonstration from the beginning of the
lesson. On an overhead or on the board, have a blank copy of the table.
Now, ask for a volunteer from each group to explain their function and to fill in the table. After
they’ve filled it in, have them use the toy-car to model the object’s path; this should make the
conjecture about the relationship between velocity, acceleration, and speed more concrete.
Position Function
Is d(t) concave up,
Velocity sign Object’s speed
down, or neither?
Acceleration sign
increasing?
d(t) =
Up
+
Yes
+
d(t) =
Down
_
Yes
_
d(t) =
Up
_
No
+
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d(t) =
Down
+
No
_
d(t) = 8t
Neither
+
No
0
Finally, ask for volunteers to answer the questions about concavity, acceleration, and speed at the
end of the worksheet.
 If the velocity function is constant, then the acceleration function is zero.
 If the position function is concave up, then acceleration is positive, and if the position
function is concave down, then acceleration is negative.
 For an object’s speed to be increasing, then the velocity and acceleration functions must
have the same signs – they should both be positive or both be negative.
Independent Practice: Students should work on the last problem of the worksheet on their own
for the remainder of the period. Before the end of class, be sure to go over the answers.
Question: The position function of an object is given by d(t) = –
.
In what intervals is velocity negative? When ‘t’ is within (- , 1) and (1,
In what intervals is acceleration negative? When ‘t’ is within (1,
).
).
In what intervals is the object’s speed increasing? Using the notion that velocity and
acceleration should have the same sign, the speed is increasing when‘t’ is within (1,
Closure: Inform the students that they’ve delved quite deeply into the concepts of acceleration,
and that they’ve seen how position, velocity, speed, and acceleration work together! They’re now
prepared to move on to other exciting applications of derivatives, and eventually 2-D kinematics.
Homework: The worksheet or textbook problems of your choice!
Created by Luke Tunstall
1-D Motion - Acceleration
Name:_______________
Student Notesheet
Warm-up: Find the inflection points and intervals of positive and negative concavity of the
position function, d(t) =
. Use a graph to support your answer. Simply looking at the
position graph, what do you think might be the significance of d’’(t)? Why?
Answer:
Acceleration is the rate at which ____________ is changing. Thus, a(t) = ______ = _______.
If velocity is measured in m/s, then acceleration would be __________.
The ________________________ of v(t) indicate possible changes in the sign of ____________.
Example: Find where the acceleration is changing if the velocity of an object is given by: v(t) =
– 4t – 2.
Groups, Part 1:
Assigned function - d(t): ____________
v(t):______________
a(t):______________
Is the position graph concave up,
concave down, or neither?
_____________________________
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For t > 0, is velocity always positive
or negative? Is the particle’s speed
increasing or decreasing?
_____________________________
_____________________________
For t > 0, is acceleration positive,
negative, or zero?
_____________________________
_____________________________
Groups, Part 2:
Position
Is d(t) concave up,
Function
down, or neither?
d(t) =
d(t) =
d(t) =
d(t) =
d(t) = 8t
Velocity sign
Object’s speed
Acceleration
increasing?
sign
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Questions to Consider:
If velocity is constant, then acceleration is ____________.
What’s the relationship between the concavity of d(t) and acceleration? ____________________
______________________________________________________________________________
Based on your observations of the sign value of velocity and acceleration, make a conjecture
about what’s necessary for an object’s speed to be increasing: ____________________________
______________________________________________________________________________
______________________________________________________________________________
Independent Practice: The position function of an object is given by d(t) = –
a. In what intervals is the velocity decreasing?
b. In what intervals is acceleration negative?
c. Using parts (a) and (b), in what intervals is the object’s speed increasing?
.
Created by Luke Tunstall
Group 1: d(t) =
Group 2: d(t) =
Group 3: d(t) =
Group 4: d(t) =
Group 5: d(t) = 8t