After understanding the role of the frames of reference, the next concepts to
be introduced in the understanding of motion are position and time.

Positions are measured in units of length, such as meter (m),
with respect to the origin of coordinates. In one dimension, to the right of the
origin of coordinates, positions are positive; while to the left of origin of
coordinates, positions are negative. The symbol for position is
x.

What is the change in position for the car in the 4
seconds represented in the drawing? The answer is 70 m. What did you
need to do to respond the previous question?

220 m -150 m =70 m

In the
animation above, the motion of the car is shown
at two different times. The initial position
of the car is identified in the drawing bythe coordinateat the
initial time,
later the car is at the coordinate
called the final position at
the final time. The relative
position between the initial and final positions are represented by the
third drawing of the sequence
.
Based on the previous example, the change in position is calculated
using the following relation
. The
change in position is the segment of the coordinate system represented
in the last drawing of the sequence.
In the same form, the change in time or time taken for this change to
occur can be calculated
using the following relation,
.

The difference between distance traveled
and displacement can be better understood
when studied in a two dimensional frame of reference. The next applet shows a
sequence of diagrams illustrating this ideas.

Suppose that, initially, there is a mobile at point A
moving toward point B, Figure I
. In this case, the mobile moves to point
B following the path illustrated in Figure II
. When the mobile is at point B, final position, how far
is from the initial position, point A? To answer this question, it is
necessary to measure the length of the straight line segment going from
point A to point B, Figure III
.
The mobile followed the longer curvilinear path rather than the straight
line path. Thus, two different measurements are derived from this
analysis; first, the distance traveled
by the mobile which is defined as the length of the path followed by the
mobile, Figure IV
. The
distance traveled can be measured by placing a string along the actual
path of the mobile and then straightening the string. On the other side,
in accordance with the definition for distance between two points, the
length of the straight line (or shortest line) from the initial
position, point A, to the final position, point B, is the
displacement (later called the
displacement vector), Figure V
. Figure VI of the
sequence shows both the distance
traveled and the displacement
. The last figure of the
sequence, Figure VII,
presents the distance traveled extended from a curvilinear line
into a straight line. This last figure allows to compare the distance
traveled with the displacement for the mobile
.

The distance traveled and the displacement are measured in
units of length. The symbol for distance traveled is
d . The displacement is the same as
the change in position.

Change in Position = Displacement

Notice that the displacement can be calculated after just to
observations of the motion. The first observation results in information about
the initial location of the mobile, initial position and the initial time. The
second observation results in information about the final position and
time for the mobile. However, for the distance traveled, it is necessary to have
a continuous number of position observations (unlimited number of observations)
in order to be able to reproduce the actual path of the mobile. This is so, even
when the time may only be measured twice. In science, it is usually expensive
and difficult to do measurements. So even when, commonly, people is more
familiar with distance traveled, displacements are simpler to work with. In
fact, just above, a relation is given for calculating displacements. There is
not a simple relation for calculating distances.

Introduction to Vectors

Distance and displacement have an additional difference,
distances are scalar quantities while displacements are vector quantities. In
this section, a brief definition of both will be given. For a more complete
description of vectors go to Vectors.

In physics, in addition to the units of the quantities,
quantities are divided in the two groups:

Physics Quantities

Scalar quantities are
defined only by their magnitude. The magnitude is the numerical value of
the quantity; example, the time of a class is 50 minutes. The magnitude
is the number 50. The unit is the minutes. Remember that except for the
case of counting objects, units are necessary for physics quantities.
Scalar quantities are usually represented by italic type, t.

Time, Mass

Distance, Speed

Vector quantities are
defined in terms of their magnitude and direction. The magnitude of a
vector is the numerical value that defines the vector just as in the
case of scalar quantities. The direction of a vector is as important as
the magnitude when stating the characteristics of a vector. The
direction of a vector on the surface of the Earth is usually defined in
terms of the cardinal points (north, south, east, and west) or the xy-plane.
Vertical directions are defined as up and down. Vector quantities are
represented by bold type, x,and/or by
.

Weight

Displacement, Velocity

In one dimension, the difference between vector quantities
and scalar quantities are minimal. The only difference is associated with the
sign of the quantity. For example, for the drawing at the top, 70 m is the
distance traveled and also the displacement. However, for the drawing below, the
distance travel still is 70 m but the displacement is -70 m. The negative sign
provides the direction for the vector. Here, the convention used is that
vectors pointing to the right are positive while
vectors pointing to the left are negative. This
convention will be use along these notes.

Exercises

The smallest displacement is obtained when an object moves from

a)

A ®
B
®
C

b)

A
®
F

c)

A ®
B
®
C ®
D
®
E
®
F

N

d)

A ®
B
®
C ®
D
®
E ®
F®
A

e)

None of the above.

The smallest distance traveled is obtained when an object moves from