Chapter 2
Describing Motion: Kinematics
in One Dimension
• Introduction
• Reference Frames and Displacement
• Average Velocity
• Instantaneous Velocity
• Acceleration
• Motion at Constant Acceleration
• Falling Objects
Introduction
Motion is part of our daily life:
 Get up in the morning
 Walk to the UofR
 Go for a lunch, etc.
If you are at rest and decide to go somewhere, you have
to “beat” your inertia and move; changing direction to
avoid a crazy teen driving a car also implies motion; etc.
Mechanics is the field in physics dedicated to the study
of motion. It is divided into two parts:
Kinematics: Describes how objects move (where an
object is at certain given time)
Dynamics: Describes the forces responsible for the
motion of objects (how they are set, and kept, in motion)
There are two types of motion: Translation and Rotation.
We will start with one-dimensional translational motion:
objects that move along a straight-line path.
(a) Translation
(b) Rotation
Reference Frames and Displacements
Suppose I say:
“A person walks toward the front of a train at 5 km/h. The train is moving at 80 km/h”.
First question we should ask ourselves: The train is at 80 Km/h with respect to what?
 Let’s assume it is with respect to the ground
Now, what is the speed of this person with respect to me if:
1) I am at rest on the train
2) I am at rest on the ground at a railway station
 5 Km/h
 85 Km/h
This example shows the importance
of identifying a frame of reference
(or reference frame) with respect to it
you can define your motion.
Ground
In our example, if you consider the person moving on the train:
- It’s speed is 5 Km/h if the reference frame is defined as the train;
- Its speed is 85 Km/h if the reference frame is defined as the ground.
Reference Frames and Displacements
In the previous example, if I ask you the question: “Is the train going from Calgary to
Vancouver at 80 Km/h with respect to the ground?”, will you be able to answer?
 No, because you do not know the direction it is moving (Calgary to Vancouver
or vice-versa?)
Despite the fact that we have identified the ground as the reference frame with
respect to it the speed of the train is measured, we have not fully characterized the
motion of the train. We also need to specify its direction.
The direction can be specified by defining a
coordinate system which represents the
reference frame used to study the motion
of an object.
In our previous example, we can choose a
point along the railroad line as the origin, or
zero “0”, of our coordinate system.
We then draw two perpendicular axes, x and y,
crossing at the origin of the system.
But, how do we use such a system to specify direction?
Reference Frames and Displacements
Definitions:
• Objects located to the right of the origin on the
x axis have positive x (+x) coordinates.
• Objects located to the left of the origin on the
x axis have negative x (-x) coordinates.
y
P
(x,y)
x
• Objects located above the origin on the y axis
have positive y (+y) coordinates.
• Objects located below the origin on the y axis
have negative y (-y) coordinates.
A point P in this coordinate system is fully identified
by a pair of coordinates (x,y)
For one-dimensional motion, we usually choose the x axis as the line of motion of an
object. If the motion is vertical, we usually choose the y axis.
Reference Frames and Displacements
+y
Some notes:
The definitions of the signs on the previous slide
are arbitrary. You can choose your own sign
convention as long as you use it consistently
while solving a problem.
+x
-x
For example:
• you could have chosen positive x positions to
be located on the left side of the origin;
-y
-y
• or have chosen positive y positions to be
located below the origin.
We will see some examples later on.
+x
-x
+y
Reference Frames and Displacements
Back to our example, let’s now locate the initial and final positions of the train in the
coordinate system we have just defined. Note that these two positions correspond to
measurements taken at two different times that we will call t1 and t2. Let’s then call
them positions x1 for t1 and x2 for t2.
Note: Positions x1 and x2 are measured relative to a point on the train (say, front of
the train) in our coordinate system.
Now we know that the train is going
from x1 to x2, so we have a direction.
Say x1 represents Calgary and x2
represents Vancouver. You are now
able to tell me which way the train is
travelling:
 Calgary to Vancouver.
x1
x2
x1 = Calgary
x2 = Vancouver
Reference Frames and Displacements
In the previous slide, the train has been displaced from its initial position x1 to its final
position x2.
We define this displacement as:
(2.1)
Note: Here the symbol Δ denotes “change”.
Note that Δx is not only a number
(scalar) but also carries information
about direction.
We can compile both information by
defining a vector:
(2.2)
Where the arrow on the top of Δx
defines a vector.
The magnitude of (2.2) is given by
(2.1).
x1
x2
Reference Frames and Displacements
Examples:
1) Suppose an object moves from position x1
measured at an initial time t1 to position x2 at a
final time t2 as shown is figure (a). The blue arrow
represents the displacement x2 – x1, or the vector
(a)
Suppose x1 = 10 m and x2 = 30 m. Then, using
(2.1):
Δx is positive denoting a vector
x direction (right direction).
in the positive
2) If we invert the initial and final positions such
that x1 = 30 m and x2 = 10 m as shown in figure
(b), then, again using (2.1):
(b)
Now Δx is negative denoting a vector
negative x direction (left direction).
in the
Reference Frames and Displacements
Note:
You shall not confuse displacement with distance travelled.
 Displacement gives how far distant an object has been displaced from its starting
point, along with the direction of such displacement. It is a vector quantity.
In the figure below, the starting position is at x1 = 0 m and the final position is at
x2 = 40 m. The displacement is:
regardless whether the object had to travel to
x3 = 70 m before reaching x2 = 40 m.
 Distance gives the total length an object
travels from its starting position, regardless
the direction of its motion. It is a scalar quantity.
The total distance is given by:
x1
x2
x3
Average Velocity
We usually make no distinction between the terms speed and velocity. However, in
physics they refer to different quantities and their distinction is very important.
Definitions:
Average Speed: Refers to how far an object travel in a given time interval and is
related to total distance.
(2.3)
Speed is a scalar (do not depend on the direction of motion).
Example: In our previous example, an object moves
from x1 = 0 m to x2 = 40 m passing by x3 = 70 m
before reaching x2. Assuming the time elapsed to go
from x1 to x2 is given by Δt = t2 – t1 = 5 s, the average
speed is:
x1
x2
x3
Average Velocity
Average Velocity: Gives not only how fast an object travels but also its direction of
motion. Contrary to average speed, average velocity is related to displacement
instead of total distance traveled. Its magnitude is given by:
or
(2.4)
Note: The bar over v in (2.4) denotes “average”.
Velocity is a vector. The sign of Δx in (2.4) defines the direction of motion. If Δx < 0,
the average velocity is negative and therefore the object is moving toward negative x
coordinates, and vice-versa.
Average Velocity
Examples:
1) Still using the example pictured in the figure (a),
an object is displaced from x1 to x2. Using
equation (2.4), the average velocity is:
x1
x2
x3
(a)
The calculated average velocity is positive, therefore the object is moving toward
positive x coordinates.
Note that, for the given example, the calculated magnitude of the average speed
(20 m/s) is bigger than that for the average velocity (8 m/s).
Average Velocity
Examples:
2) The position of a runner as a function of time is plotted as moving along the x axis
of a coordinate system. During a 3.00 s time interval, the runner’s position
changes from x1 = 50 m to x2 = 30.5 m, as shown in figure (b). What was the
runner’s average speed? And its average velocity?
Average velocity:
(b)
Note: The absolute magnitude of both the average speed and average velocity
are the same. This is always true when the motion is all in one direction.
Instantaneous Velocity
So far we have used the term “average” for velocity and speed. This is due to the fact
that the velocity can change along the path of motion of an object, though we can
always assign an average velocity (or speed) to it knowing the time elapsed for the
object to go from an initial to a final position.
For example:
a) constant velocity.
 Figure (a) depicts a car moving at constant
velocity. In this particular case, the average
velocity is similar to the velocity of the car
at any instant of time.
 On the other, figure (b) represents a car moving
at different velocities at different instant of time. In
this case, the average velocity is not necessarily
equal to the car velocity at a particular instant of
time.
(b) varying velocity.
Instantaneous Velocity
We can use eq. (2.4) to define the velocity at any instant of time, or as we call:
instantaneous velocity. For this purpose, we can make the time interval very short.
In fact, we can make it so small that it will be close to zero. We say that Δt is tending
to zero, or that it is an infinitesimally short interval of time.
(2.5)
a) constant velocity.
Note that I have dropped the bar on the top of v.
So, the magnitude of the instantaneous velocity
is represented by v in (2.5).
From now on, every time I use the term velocity
it will refer to the instantaneous velocity v instead
of the average velocity, unless otherwise stated.
(b) varying velocity.
Δx
Acceleration
 Acceleration gives how fast the velocity of an object is changing.
The average acceleration is defined as the change in velocity divided by the time
taken to make this change:
(2.6)
As velocity, acceleration is also a vector. It can have either positive or negative sign:
 If v2 > v1 , Δv > 0 implies that the acceleration vector points towards positive x axis
 If v2 < v1 , Δv < 0 implies that the acceleration vector points toward negative x axis
Note that acceleration has the units (SI) of m/s2.
Acceleration
An object can go faster (accelerate) or slow down (decelerate) depending on the
relative direction between the velocity and acceleration vectors.
The two examples below illustrated these concepts:
1) An car moves to the right of a coordinate system as shown in figure (a) (toward
positive x axis). The car slows down from an initial velocity v1 = 15.0 m/s to a final
velocity of v2 = 5.0 m/s in a time interval of 5.0 s. What is the car’s average
acceleration?
Using eq. 2.6,
Note that the acceleration is negative and points
in the opposite direction of the velocity.
Note also that if we invert the initial and final velocities such now v1 = 5.0 m/s and
v2 = 5.0 m/s, the resulting average acceleration will have the same magnitude but
with a positive sign pointing in the same direction as the velocity vector. Therefore, in
this case we have an acceleration.
Acceleration
Example: In this figure, the car is accelerated from a velocity v1 = 0 Km/h to v2 = 75
Km/h at a rate of 15 Km/h per second.
Acceleration
Similarly to instantaneous velocity, we can define instantaneous acceleration as:
(2.7)
.