Created by Luke Tunstall
Position and Velocity – 1-D Motion for Calculus Students (90 Minutes)
Learning Goals: Using graphs and functions, students will explore the relationship between
motion on a line and a position function. Students will also examine the interplay between
position, velocity, and speed.
Prior Knowledge: 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.
2.06: 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:
 Individual whiteboards for each student
 Notesheet for each student
Think-Pair-Share Warm-up: Students should work on the warm-up by themselves for a few
minutes, and then discuss their answers with a partner. Following the partner discussions, ask for
a duo to come up to the board to share their answer and solution.
Question: If f(x) = 3
– 6, find the intervals in which f(x) increasing or decreasing.
Answer: We see that f(x) has a critical point at x = 2, and so it is neither increasing nor
decreasing at x = 2. Because f’(x) = 9
, it must be the case that f(x) is decreasing from
(- , 2) and from (2, ).
Following the warm-up, introduce the students to 1-D kinematics by explaining that they’ll use
what they know about derivatives and critical points to examine the motion of objects, rather
than just when a function (like f(x) from the warm-up) is increasing and decreasing. If we can
model objects along a straight line, then we can consider situations like a bullet fired straight in
the air, a person jumping out of a plane, or a particle falling in water.
Instructional Activities:
Lecture: Draw a straight line on the board, and explain that if an object is moving along the line
in time, we seek a way to model such motion.
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Ask the class if anyone has any ideas for how to model before telling them.
In the case that no one has the right idea, inform the students that there are two ways to model
motion on a straight line: They can directly observe the motion, marking the positions at
different times, OR they can make a graph of time versus position with respect to the origin.
Such a graph will represent a position function, which we’ll call d(t). If we have a position
function, we simply input a time ‘t’ and our output tells us where the object is. Draw the
following the number line on the board to illustrate what we mean by a line with an origin. Make
sure it’s above your height, as you’ll be using it later.
Demonstration: Now, tell the students it’s time to see how motion along the line translates to
the graph of time versus position. Hand out the whiteboards to each student. Make sure that the
front whiteboard still has the number line. On the students’ whiteboards, they should draw an
empty graph of time versus position. They’ll fill it in according to how you walk:
Demo 1:
 Begin 5 feet to the right of the origin.
 Walk 10 feet (at a steady pace) to the left so you end up 5 feet to the left of the origin.
 Stay still at the spot for a few seconds.
 Walk 7 feet (at a steady pace) to the right so you end up 2 feet to the right of the origin.
 Stay still at the spot for a few seconds.
Give the students a minute, and then have them hold up the whiteboards to check for
understanding. The position graph should look something like this:
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Demo 2:
 Begin at the origin.
 Walk 5 feet to the right (at a steady pace).
 Stay still at the spot for a few seconds.
 Sprint for 10 feet to the left so you end up 5 feet to the left of the origin.
 Stay still at the spot for a few seconds.
 Walk 3 feet to the left (at a steady pace) so you end up 8 feet to the left of the origin.
Give the students a minute, and then check their whiteboards again for understanding. Be sure
that their slope is steeper for the sprint! The position graph should look as follows:
Now that the students have been introduced to position graphs, it’s time that they further explore
the relationship between position and slope, as this will serve as a bridge to velocity.
Group Activity: Break out the students into groups of 3 or 4, and have them go through the
corresponding “Groups” section of the worksheet. They’ll look at three position functions and
answer questions about slope – within each group, encourage one member to look at one graph.
As the students are working and discussing, the teacher should walk around as the students are
working to see if they have any questions.
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After the students have had about 20 minutes, ask for different individuals from different groups
to explain their answers for the position graphs. The students should begin to see that slope of the
position function directly relates to the direction of motion. The following is a short key to the
section of the worksheet:
 Graph 1: The object is moving to the right, takes a break, and continues moving to the
right. The corresponding slope is positive, zero, and then positive.
 Graph 2: The object is moving to the left at a constant rate. The corresponding slope is
negative.
 Graph 3: The object is moving to the right, taking a break, and then moving to the left.
The corresponding slope is positive, zero, and then negative.
Lecture: The students should see from the group-work that the slope of the position function is
related to the object’s movement. The students should separate from their groups and take the
following notes on their notesheet. The underlined portions correspond to the empty parts of the
students’ notes.
(Before writing each concept on the board, ask the students for input so that they shout out their
own ideas)
Velocity is defined as the rate at which position changes in time. Because it is dependent upon
which direction the object is moving, it can be both positive and negative.
(Explain that because slope is a rate, we can take the derivative of position to obtain velocity)
Theorem: v(t) = d’(t).
The critical points of d(t), when v(t) = 0, indicate a possible change in the sign of velocity. This
is because the object is at rest.
(Write the following example on the board, give them a few minutes, and then have the students
show their answer on their whiteboards)
Example: Find where the velocity is changing if the position function of an object is given by
d(t) = 6 – 10t.
Answer: The velocity switches from negative to positive when t = (5/6).
Speed is always positive, and is the magnitude of velocity. Thus, s(t) = |v(t)|
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Example: Draw the following velocity graph on the board. On the students’ whiteboards, have
them draw the corresponding speed graph. After a minute, have them raise the boards to check
for understanding.
Independent Practice: Now that the students have covered all the material from the lesson, for
the remainder of the class they should work on the following practice problems which are on the
worksheet. Walk around to offer assistance.
Draw an arrow from the position or velocity functions to the correct statement!
d(t) =
– 2t + 2
v(t) =
– 16t + 64
d(t) = 6
Object at permanent rest
(8,
) Moving to the right
(0, 8) and (8,
) Moving to the right
Closure: Explain to the students that they’ve learned a lot today, but that velocity isn’t the only
factor that relates to motion. Ask the students if they have any guesses for what else might be
important – especially if they consider cars. Someone will likely bring up acceleration, so tell the
students that they’ll learn about how acceleration relates to position, velocity, and speed soon.
Homework: Assign the worksheet or textbook problems of your choice.
Created by Luke Tunstall
One-Dimensional Kinematics – Position and Velocity
Student Note-Sheet
Name: __________________
– 6, in what intervals is f(x) increasing? Decreasing?
Partner Warm-up: If f(x) = 3
Neither?
There are two ways of representing motion along a line. Write or draw a description of each.
1.
2.
Group Activity:
How is the object moving? ____________________________________
___________________________________________________________
How does the movement correspond to the slope? _________________
______________________________________________________
_____
How is the object moving? ____________________________________
___________________________________________________________
How does the movement correspond to the slope? _________________
______________________________________________________
_____
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How is the object moving? ____________________________________
___________________________________________________________
How does the movement correspond to the slope? _________________
______________________________________________________
_____
Based on your observations, what can you determine from the slope of the position graph?
______________________________________________________________________________
Notes:
Velocity is defined as the rate at which ___________ changes in time. Because it is dependent
upon which direction the object is moving, it can be both ____________ and ___________.
Theorem: v(t) = ______
The critical points of d(t), when v(t) = 0, indicate a possible change in ________. This is
because the object is at _________.
Example: Find where the velocity is changing if the position function of an object is given by
d(t) = 6 – 10t.
Speed is always ____________, and is the magnitude of velocity. Thus, s(t) = _____.
Example:
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Solo Practice:
Draw an arrow from the position or velocity functions to the correct statement!
d(t) =
– 2t + 2
Object at permanent rest
v(t) =
– 16t + 64
(8, ) Moving to the right
d(t) = 6
(0, 8) and (8,
) Moving to the right