Chapter 3: Introduction to Kinematics – The Constant Velocity Particle Model (CVP)
Kinematics is the study of motion. In this chapter we will study constant velocity in one dimension. That means
the object is moving with the same speed in a straight line. The line may be horizontal, vertical, or at an angle.
When we call an object a particle, it means that the size or composition of the object is not an important factor in
its behavior. In this unit we will build a model for motion of a particle at a constant velocity using three
representations. You will learn to use graphical representations, motion maps and mathematical equations.
Part 1: Vocabulary of motion
Here is a quick look at some of the vocabulary of motion.
Position: the location of an object. The variable for position is the letter “x”. To measure a position, we pick a
spot, call it the origin (zero), and measure the distance from this spot to the object. Zero position may be
anyplace one wants it to be.
Distance: the length of the path an object moves. The variable for distance is the letter “d”.
Displacement: The change in position of an object between two points. To find the displacement, the straightline distance from the starting position to the final position is measured. The direction this line points is the
direction of the displacement. The variable for change in position is “x”. The symbol ““ is pronounced delta,
and stands for “change in”. By definition, displacement is final position “xf” minus starting position “xI” or in the
form of an equation x  x f  x i .
Distance is a scalar quantity. Scalar quantities have only magnitude (an amount), direction is not important.
Displacement is a vector quantity. Vector quantities have both magnitude and direction. In this chapter, since
the objects are always moving in a straight line, we use directions like up and down, right and left, north and
south, etc. The standard unit of distance and displacement is the meter (m), although many other units are
used.
Speed: how fast an object is moving or the rate of change in distance. The variable for speed is the letter “v”.
Velocity: how fast and in what direction an object is moving or the rate of change in displacement. The variable
for velocity is also the letter “v”.
Most people use the terms velocity and speed interchangeably. To a physicist, they are different in that velocity
has direction associated with it and speed does not. Velocity is a vector, and speed is a scalar. The direction of
the object’s velocity is the same as its displacement.
The definitions of speed and velocity are: average speed 
distance traveled
d
or v 
elapsed time
t
average velocity 
change in position
x x f  x i
or v 

elapsed time
t
t f  ti
“tf” stands for final time, and “ti” for initial time. The bar over the v stands for average. The standard unit for time
is the second (s), and for both speed and velocity is meters per second (m/s). Unless the units are specified in
the problem, any units of length and time may be used.
Part 2: Graphical and mathematical models
position versus time
Lets look at the graph to the right to see what we can find out about
the motion of the object moving at a constant velocity.
a) What is the object’s velocity?
Velocity is the change in position divided by the change in time. So
velocity is the slope of a position/time graph.
x f  3.3
s
t  10m
c) What is the general equation for the position of an object
moving at a constant velocity?
x f  v t  xi
final position  velocity  time  initial position
50
40
X (m)
x 50m  10m 40m


 3.3 ms
t
12s  0
12s
b) What is the equation (mathematical model) for this object’s
position?
m
v
60
30
20
10
0
0
2
4
6
t (s)
8
10
12
19
a) At the 3rd second, the slope of the line is:
position (m)
The basic principles of motion graphs can best be shown by working out a few sample problems. You will get
both quantitative (using numbers) and qualitative (correct general shape) graphing problems in this unit.
Sample problems:
12
position versus time
1) Using the graph to the right, calculate the object’s velocity at
a) 3 s? b) 10 s?, c) 20 s? d) What are the average velocity
10
and the average speed for the trip? e) Sketch v/t graph for
the trip.
8
Solution:
(Locate the point for that time and calculate the slope of the
6
line on which the point is located.)
x 8m - 2m

 1.2 ms
t
5s - 0
4
b) At 10 s the line is horizontal so the slope is 0, the object is not
moving.
2
c) At the 20th second, the slope is: x  0 - 8m  .44 ms (negative
0
0
t
30s - 15s
5
10 15
time (s)
shows object is traveling backwards.)
d) For the average velocity the use the displacement for the trip.
25
30
x 0m - 2m

 .067 ms
t
30s - 0
For the average speed the use the distance moved during the trip.
e)
v
20
d 6m  8m

 .47 ms
t
30s
15
velocity vs. time
t
i) For 0-2 s, area = 10m/s x 2s = 20.m
ii) For 2-5 s, area = -10m/s x 3.0s = -30.m
(Negative when the line is below zero.)
iii) The last 1 s is 5 m/s x 1 s = 5.0 m
b) What is the displacement for the whole trip?
Whole trip is (20 m + -30 m + 5 m) = -5.0 m.
c) When is the object at the same position as zero time?
When the displacement for the trip is zero, which means
the object is back where it started. This happens at t =
4.0 s. In the first 2 seconds it traveled +20 m and in the
next 2 s it traveled –20 m.
d) Construct a quantitative position/time graph for the trip.
The velocity/time graph does not let us know the starting
position, so the position 10 m is chosen. Then the object
moved 20 m forward, 30 m backward and 5 m forward.
The object ends up 5 m behind the starting point, which
agrees with the velocity/time graph. Also shown is that
the object is back at the starting position at time 4 s.
20
5
0
-5
-10
-15
0
1
2
3
4
5
6
time (s)
30
position versus time
25
20
position (m)
2) Answer the following questions using the velocity/time graph to
the right.
a) What was the object’s displacement in 0-2 s?,2-5 s? 5-6 s?
(The displacement is the area of the graph between the
x-axis and the line, since x = vt.)
Velocity (m/s)
10
15
10
5
0
0
1
2
3
time (s)
4
5
6
Below is a summary of the information one can gather using the lines on constant velocity graphs.
position/time graphs
1. A horizontal line or section indicates the object is at rest.
2. A straight line at an angle represents motion at a constant speed. The steeper the line the faster the motion.
3. Upward above zero represents motion away from the origin. Downward, toward the zero represents motion
back toward the origin.
4. The slope of a line on a position/time graph is the velocity of the object.
velocity/time graphs
1. A horizontal section or line represents motion at a constant velocity.
a) on the x-axis means the object is not moving.
b) the farther above or below the axis, the faster the object is moving.
2. If the line is above the zero, the object is moving away from the origin. If the line is below the zero, the object
is moving toward the origin.
3. The area between the motion line and the x-axis represents the object’s displacement.
Part 3: Motion maps
A motion map can be used to represent the position, velocity, and acceleration of an object at various clock
readings. (At this stage of the course, you will be representing position and velocity only.) This section is to
explain how motion maps are used to represent constant velocity.
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.
Below is the motion map that represents the car. We indicate the position of the object with a small point.
At each position, the object's velocity is represented by a velocity arrow. The arrows show the magnitude of
the velocity with their length, and the direction with the way they point. Quantities that have both magnitude
(a numerical value) and direction are called vectors. The arrow and the + are to show which direction is
positive for position. This is also the positive direction for velocity.
+
If the car is 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,
indicating that the car is moving faster.
+
If a car is moving to the left at constant velocity, the photo and motion map might look like this:
+
More complicated motion can be represented as well. When making motion maps, if position dots would be
placed on top of each other if you stay on the same level, place one above the other.
+
Here, an object moves to the right at constant speed for three time intervals, stops and remains in place for two
time intervals, then moves to the left at a slower constant speed for three time intervals.
21
Sample problem:
3) Below are the motion maps of two runners.
a) Describe the motion of each runner
b) If each time interval is ten seconds, about when do they meet?
c) Draw a graphical representation of the runs. Put both runners on the same graph.
+
B
A
a) At time t = 0, runner A starts at a position to the right of zero position and moves to the right at a
constant velocity. Runner B starts at the origin and travels to the right at a constant velocity that is
greater than A’s.
b) At a time a little before t = 30 s, probably around 28 s B
catches A. (B’s third dot, which is the 20 s dot, is behind
A’s third dot. However B’s fourth dot, the 30 s dot, is
slightly ahead of A’s fourth dot.)
x
A
c) A graphical representation of the behavior of runners
A and B could like this. (The line for B is steeper
because B is faster. A started ahead of B so the A
line starts higher. Where the lines cross, the runners
are even.
Part 4: Problem solving kinematics story problems
B
0
10
20
30
t (s)
These three equations and an understanding of them will allow you to solve all story problems involving constant
velocity. d  vt, x  vt, x f  vt  x i The last two equations are actually the same. The last is used when
solving for an actual position, and the middle one is used when solving for a change in position.
Problem solving strategy
1. Draw a simple sketch of the object’s motion. Use arrows to show the path of the motion. Some students
also draw a graph of the motion to help them solve problems.
2. Choose which direction will be positive.
3. List your known and unknown quantities in the problem, including the units.
4. Check to see that the units are consistent.
5. Pick the kinematics equation/s, which will enable you to determine the unknowns.
6. Put in the numbers with units and solve. Cancel out any units you can.
7. Check to see if the final answer is reasonable, and that the final unit is appropriate.
Sample problem:
4) Marty Graw is out for a bike ride. He travels west at 12.0 m/s for 20.0 minutes and then travels east at
10.0 m/s for 30.0 minutes. a) What distance did he travel? b) What was his displacement? c) What was
his average speed? d) What was his average velocity?
Equation: x = v t
west
+
then east
-
Knowns: (part 1 of trip)
v =12m/s
t = 20 min = 1200s
Knowns: (part 2 of trip)
v =-10m/s
t = 30 min = 1800s
Unknowns: d and x
22
Solution:
Part 1: Δx  12 ms (1200s)  14,400m
Part 2:
Δx  10 ms (1800s)  18,000m
a) distance = 14,400m + 18,000m = 32,400m
(sign doesn’t matter for distance)
b) displ = 14,400m + -18,000m = -3,600m or 3600m east
c)
v
32400m
3000s
 10.8 ms
d)
v
-3600m
3000s
 1.2 ms
Problems:
1) Below is a position vs. time graph for June Niper and Cam Ralenns as they ride their bikes.
Ju
ne
a) Do the cyclists start at the same point? How do you know?
If not, which is ahead?
x (m)
Cam
5s
t (s)
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?
d) Are their velocities equal at any time? How do you know?
e) What is happening at the intersection of the two lines? How do you know?
2) Below is a position vs. time graph below for Phil Angese and Tim Tation as they ride their bikes.
a) How does the motion of Phil in this graph compare to that
of Cam above? (assume same scale)
x (m)
Tim
Ph i l
5s
b) How does the motion of Tim in this graph compare to that of
June in the top graph? (assume same scale)
t (s)
23
c) Which cyclist has the greater speed? How do you know?
d) Describe what is happening at the intersection of Tim and Phil’s lines.
e) Which cyclist has traveled further during the first 5 seconds? How do you know?
3) Bill Melater, roller skating down a marked sidewalk, was observed to be at the following positions at the times
listed below:
t (s)
x (m)
0.0
4.0
1.0
7.0
2.0
10.0
5.0
19.0
8.0
28.0
10.0
34.0
a) Plot a position vs. time graph for Bill.
b) What is Bill’s average speed for the trip?
c) Write the mathematical model (equation) to describe the line in (a).
d) Based on your model in (c) at what position will Bill be at t = 16 seconds?
e) How much time will it take Bill to travel 100.m?
24
4) When Ann Thropology skated, the following data was taken.
a) Plot the position vs. time graph for Ann.
t (s)
x (m)
0.0
0.0
2.0
6.0
4.0
12.0
6.0
12.0
8.0
8.0
10.0
4.0
12.0
0.0
b) What do you think is happening during the time
interval t = 4 s to t = 6 s? How do you know?
c) What is Ann’s velocity at time = 7 s?
d) Determine Ann’s average speed from t = 0s to t = 12s.
e) Determine Ann’s average velocity from time t = 0 to time t = 12 s.
f) Draw a motion map for the trip where the strobe is flashing at each second clock reading.
g) What is Ann’s displacement for time t = 7 s to t = 11 s?
h) Plot a graph of velocity vs time for her trip.
i) Cross hatch the area above the x-axis and below
the line on the graph. Calculate the value for this
area. What does this area represent?
0
crosshatching
25
j) Calculate the area underneath the x-axis and above the line on the graph. What does this area represent?
k) Add the areas from (i) and (j) together. What does this value represent?
5) Sketch the position vs time and the velocity vs time graphs corresponding to the following descriptions of the
motion of an object.
a) Ella Vader moves away from the 10 m position
at a constant (steady) speed.
v
X
(m)
( ms )
t (s)
t (s)
b) Matt Respad moves toward the zero position at
a steady speed for 10 s, then stands still for 10
s.
X
v
(m)
( ms )
t (s)
t (s)
c) Tor P. Doe moves away from the zero position
at a steady speed for 10s, reverses direction and
moves back toward the zero position at the
same speed.
X
v
(m)
( ms )
t (s)
t (s)
6) In this section fill in the 3 empty boxes that represent the same motion as the one shown. On the motion map
have a dot for each vertical dashed line on the graphs.
x vs. t Graph
v vs. t Graph
Written Description
a)
v
( ms )
x
(m)
t (s)
26
t (s)
Motion Map
x vs. t Graph
v vs. t Graph
Written Description
Motion Map
b)
v
( ms )
x
(m)
t (s)
t (s)
c)
v
( ms )
x
(m)
t (s)
d)
x
(m)
v
( ms )
t (s)
t (s)
e)
x
(m)
v
( ms )
Object moves with
constant positive velocity
for 4 seconds. Then, it
stops
for 2 seconds and
t (s)
returns to the initial
position in 2 seconds.
t (s)
f)
x
v
( ms )
(m)
t (s)
+
t (s)
27
g)
Object A starts 10m to the
right of zero position and
moves to the left at 2 m/s.
x
(m)
v
( ms )
t (s)
Object B starts at zero
position and moves to the
right at 3m/s.
t (s)
h)
(Both objects started at
the same time.)
x
(m)
v
( ms )
A
t (s)
+
B
t (s)
i)
(Both objects started at
the same time.)
x
(m)
v
( ms )
B
t (s)
A
t (s)
15
10
v (m/s)
7a) The motion of Ms. Teri Novel is shown on the
graph to the right again at the same position as
at time = 0 s? How do you know?
+
b) What is the her displacement for the time shown on
the graph?
5
0
-5
5
-10
10
-15
0
c) What is her average speed for the trip?
d) What is her average velocity for the trip?
e) On the grid to the right, make a quantitative graph
for position vs time.
28
2
t (s)
4
6
8) Sue D. Nymm bikes east 200. m to her friend’s house at 10.0 m/s. She takes 5.0 s to drop off her Physics
data. She then walks 120. m west at 3.0 m/s. What was her:
a) trip distance?
b) trip displacement?
c) average speed?
d) average velocity?
9) Justin Credible runs east from zero position at 5.0 m/s. Ann Gelic is running east at 3.0 m/s, and at zero clock
reading is 8.0 meters ahead of Justin. How much time does it take Justin to catch her and at what position
are they? (Solve both graphically and mathematically.)
10) Leah Tard is riding her bike north at 10.0 m/s and Willy Makit is traveling south at 8.0 m/s. At time = 0, Willy
is 500. m away from Leah. How much time does it take until they meet? What is the displacement of each
cyclist at the time they meet?
29
Additional Problems:
Sketch in the corresponding graphs that go with the graph already given to you.
1a)
1b)
1c)
x
x
x
t
t
1a)
t
1b)
v
t
1c)
v
t
1d)
v
t
1e)
v
x
v
x
t
t
t
t
6
2) To the right is a graph of Larry Ett’s position for a six second trip.
a) What is his velocity at 1 s?, 4 s?, 5.5 s?
position versus time
5
c) What is his average speed for the trip?
4
position (m)
b) What is his average velocity for the trip?
3
2
1
d) Sketch a velocity/time graph of the trip.
0
0
3a) What is Neil Down’s displacement for the trip to the right?
1
30
2
3
time (s)
4
5
6
velocity vs. time
20
c) What was his average velocity for the trip?
d) Draw a quantitative
position/time graph for his trip.
velocity (m/s)
b) At what time did Neil return to the start?
10
0
-10
-20
-30
0
30
5
10
15
time (s)
20
25
30