Lecture 2.1
Kinematics in One Dimension
Displacement, Velocity and Acceleration
Everything in the world is moving. Nothing stays still. Motion occurs at all scales
in the universe, starting from the motion of electrons in atoms, continuing with thermal
motion of molecules and ending with motion of stars and galaxies at the scale of the
universe itself. Classical mechanics studies motion of objects at every-day scale. Even
though classical mechanics requires corrections to describe motion at very small, atomic,
scale as well as at very large scale of the universe, but it still serves as the basis for
description of motion at these levels too.
Think what does it mean to study motion? What are the basic steps in this process,
what sort of questions one has to answer about motion?
Classical mechanics can be further subdivided into several parts. The part of
classical mechanics which deals with classification and comparison of motion called
kinematics (from the Greek word kinema, meaning "motion"). The main question of
kinematics is “How does the motion happen?” However, we cannot study motion without
understanding of its reasons. This is why mechanics is not limited by kinematics only.
The other part of mechanics, which studies causes of motion and deals with question
“Why does the motion happen?” is called dynamics. Dynamics is based on the three
Newton’s laws. We will discuss these laws later during the semester. Now we shall start
our study of kinematics.
The description of motion is sometimes difficult and challenging task. You have
just seen that while performing the lab exercise. Since physics is quantitative science, we
shall introduce quantitative characteristics of motion in such a way that they can be
measured experimentally. (We shall measure these quantities in the lab very soon). Then
we shall examine relations between those characteristics.
For today we shall only restrict our attention by motion along the straight line (onedimensional motion). However, this motion can be in any direction. It could be a car
moving along the straight part of the highway, or it could be King Kong falling straight
down from the top of the Empire State Building. In both cases we will only consider
objects moving as a whole. This means that the motion of such an object is very much
like the motion of a particle. All the parts of the object are moving in the same direction
and at the same rate. King Kong is just falling down without any rotation around his own
axis and every portion of the car moves in a same way.
1. Position, Distance and Displacement
To describe motion first we have to locate the object in space. This can be done
only relative to some reference point. The choice of this reference point depends on the
problem. For instance, for the car it can be the place where it started moving, for King
Kong it can be either the top of the Empire State Building or the ground level. The
reference point serves as the origin (zero point) of the coordinate system. Since today we
are talking about one-dimensional motion, the coordinate system only has one axis. Then
we have to choose positive direction in which coordinate will increase. This choice
depends on our decision. For my first example, it is natural, but not required to choose
positive direction to be the same as direction of the car's motion. But even if the car turns
back after traveling some time, we still have to keep the same positive direction of the
coordinate axis; we cannot change the rules during the game. It seems that in the example
about King Kong, it does not make too much difference to have positive direction either
upwards or downwards.
So now we can define position of the object by its coordinate. If we call the axis to
be x-axis, then position of the object is defined by its x-coordinate. If the object moves
from its position at some x1 to another position at x2 , we shall call the change of its
position x displacement
x  x2  x1
(2.1.1)
If an object moves in the positive direction of axis x then x2  x1 and displacement, x ,
is also positive. If it moves in the negative direction then x2  x1 and its displacement is
negative. So displacement has not just the magnitude x but also it has the direction.
Physical quantities which have both magnitude and direction are called vectors. This
means that displacement is a vector. In fact, displacement only depends on the final and
original position of the object but not on the distance the object has traveled. For
instance, if a car goes from Abilene to Dallas and then back to the same place in Abilene,
its displacement is zero, while it traveled quite a long way. In the case of the motion
along the straight line direction of displacement can be shown by its sign. In general,
however, it is not enough to say that displacement is positive or negative. Moreover, you
can not even say that, if the displacement vector is not directed along the same axis of the
coordinate system.
In contrast with displacement, the distance traveled is equal to the total length of
the path traveled and it does not depend on the final and original positions but rather on
the path itself. It is a scalar. However, it is also measured in units of length.
2. Average Velocity and Average Speed
Our reference frame is not completed yet. This is because every event occurs not
only somewhere in space but also somewhere in time. So, to obtain the full physical
x (m)
t (s)
description of motion, we have to upgrade our coordinate system by adding a measuring
device to count time t. Again, we have to choose a reference point, the original moment
in time, when we start our observation. In my examples it can be the moment, when the
car starts from rest or when King Kong starts falling down from the top of the building.
Now we can see not only how far the object goes, but also how fast it is. This can be
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presented graphically as the dependence of the position on time x t
Here, we have a graph for the motion of a car. It starts from the original position
x0  1600m at time t0  0 and then it moves along the x- axis in the positive x-
direction. At time t  40s , it crosses the origin on the coordinate axis and continues to
move in the same direction afterwards.
Think about various possibilities for the graphs of position vs. time.
To understand how fast this car is moving one can introduce several quantities. The
first will be the average velocity which is the ratio of the displacement x to the time
interval t for which this displacement occurs
vavg 
x x2  x1

.
t
t2  t1
(2.1.2)
As you can see the average velocity depends on the time interval for which it is found.
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Since the x t -dependence is, generally speaking, a curve, not a straight line, the average
velocity will be different for every different time interval. It is, in fact, the slope of the
straight line connecting any two points of the curve. Different points provide different
straight lines with different slopes. As we can see from its definition, the average velocity
always has the same sign as the displacement x . So, it has certain direction and it is a
vector in a same way as displacement. If displacement is equal to zero, then the average
velocity is zero too. For a car, which after a long trip has finally come back to the same
point where it started, the average velocity is zero. It does not matter, how fast the car
was moving during its trip. This does not provide enough information. So, along with
average velocity we shall also consider an average speed, which is defined as the ratio of
the total distance traveled (does not matter in which direction) for the certain time
interval to this time interval
savg 
total distance
.
t
(2.1.3)
The average speed, in contrast to average velocity, does not have any sign or direction, so
it is a scalar, but it still depends on the time interval for which it is taken.
3. Instantaneous Velocity and Speed
Average velocity as well as average speed refers to some time interval. However,
for the most part, we want to know how fast something is moving at a given instant. To
see that, we have to shrink the time interval in definition of average velocity to zero,
which is
v  lim
t 0
x dv
,

t dt
(2.1.4)
called instantaneous velocity. So, instantaneous velocity is the first derivative of the
position with respect to time. Graphically instantaneous velocity represents the slope of
the curve in your graph at any given moment in time. Instantaneous velocity is also a
vector, having both magnitude and direction. At the same time, the magnitude of
instantaneous velocity is called instantaneous speed. Instantaneous speed is the quantity
shown by the car's speedometer.
Example 2.1.1. Suppose a car passes through 10 miles construction zone at 20 mph
and then travels at 60 mph for another 10 miles. What is the average velocity of the car
during that time?
Work through this example before looking at the solution.
In order to be able to use the definition of the average velocity 2.1.2, we first have
to find time it takes for this car to pass that distance. It passes first 10 mi for time
t1 
x1
10mi
1

 hr ,
v1
20mph 2
and second ten miles for time
t2 
x2
10mi
1

 hr .
v2
60mph 6
So the average velocity will be
vavg 
x x1  x2
10mi  10mi
20mi



 30mph .
t t1  t2 1 2 hr  1 6 hr 4 6 hr
The common mistake for this problem is to calculate the average of two velocities 40
mph instead of average velocity 30mph.
Exercise Prove that the average velocity equals to the average of the two velocities
b
g
v  v1  v2 2 only if the car moves the same time with each of these two velocities.
Example 2.1.2. Position of the toy car is given by the equation x  20t  5.0t 3 ,
where x is measured in cm and time, t, in s. At what time will this car have a zero
velocity?
The velocity of this car can be found as
v
dx d
  20t  5.0t 3   20  15t 2 .
dt dt
If this velocity becomes zero, one has
20  15t 2  0,
15t 2  20,
t2 
20 4
 ,
15 3
t
4
2

 1.2
3
3
So, this velocity will be zero at 1.2 seconds.
Exercise: Graph position and velocity of this car as functions of time.
4. Acceleration
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If you look again at the x t -graph, you can see that it has a different slope at every
point. In fact, in the picture above this slope gets larger with time, this means that the car
is not moving with constant velocity, but accelerates all the time. This is what happens,
when the driver presses the accelerator pedal. To see how fast velocity of the car is
changing during the certain time interval, one can introduce the average acceleration in a
same way as we have done for the average velocity:
aavg 
v2  v1 v
.

t2  t1 t
(2.1.5)
Average acceleration depends on the time interval for which it is calculated. It is a vector.
On the other hand acceleration and velocity are quite different things, since the later
represents how fast the object is moving while the former represents how fast its velocity
changes.
The instantaneous acceleration is the acceleration at a given moment in time
v dv d 2 x
.


t 0 t
dt dt 2
a  lim
(2.1.6)
Instantaneous acceleration is the first derivative from instantaneous velocity with respect
to time or the second derivative from position with respect to time. The instantaneous
acceleration can be seen as the slope of the curve, which depicts dependence of the
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velocity on time, v t .
As it can be seen from definitions, since the SI unit for velocity and speed is m/s,
the SI unit for acceleration is m s2 .
Example 2.1.3. Let us continue with the same example. The position of the toy car
is given by the equation x  20t  5.0t 3 , where x is measured in cm and time in s. When
this car will have zero acceleration? When will it have negative and when positive
acceleration?
The velocity of this car was found as
v
dx d
  20t  5.0t 3   20  15t 2 .
dt dt
To find car’s acceleration, let us use
a
d
i
dv d

20  15t 2  30t .
dt dt
So the only time, when this car has zero acceleration is at the very beginning of its
motion at t=0. After that it always has negative acceleration.