Lesson 3 : Describing Motion
with Position vs. Time Graphs
The Meaning of Shape for a p-t
Graph
The study of 1-dimensional kinematics has been
concerned with the multiple means by which the motion of
objects can be represented. Such means include the use of
words, the use of diagrams, the use of numbers, the use
of equations, and the use of graphs. Lesson 3 focuses on
the use of position vs. time graphs to describe motion.
The specific features of the motion
of objects are demonstrated by the shape and the slope of the lines on a
position vs. time graph. The first part of this lesson
involves the study of the relationship between the motion of an object
and the shape of its p-t graph.
To begin, consider a car moving with a
constant, rightward (+)
velocity of 10 m/s.
If
the position-time data for such a car were graphed,
the resulting graph would look like the graph at the
right. Note that a motion with constant,
positive velocity results in a line of constant and
positive slope when plotted as a position-time graph.
Now consider a car
moving with a changing, rightward (+)
velocity – that is, a car that is
moving rightward and speeding up or
accelerating.
If the position-time data for such a car were graphed,
the resulting graph would look like the graph at the
right. Note that a motion with changing,
positive velocity results in a line of changing and
positive slope when plotted as a position-time graph.
The position vs. time graphs for the
two types of motion – constant velocity and changing
velocity (acceleration) – are
depicted as follows:
The shapes of the position vs. time
graphs for these two basic types of motion – constant
velocity motion and changing velocity motion (i.e.accelerated motion)
– reveal an important principle.
The principle
is that the slope of the line on a position-time graph
reveals useful information about the velocity of the
object. It's often said, "As the slope goes, so goes the
velocity."
Whatever characteristics the velocity has, the
slope will exhibit the same (and vice versa). If the
velocity is constant, then the slope is constant (i.e., a
straight line). If the velocity is changing, then the
slope is changing (i.e., a curved line). If the velocity
is positive, then the slope is positive (i.e., moving
upwards and to the right). This principle can be
extended to any motion conceivable.
Example 1
Consider
the graphs below as examples of this
principle concerning the slope of the line on a position
vs. time graph.
The graph on the left, below, is representative
of an object which is moving with a positive velocity (as
denoted by the positive slope), a constant velocity (as
denoted by the constant slope), and a small velocity (as
denoted by the small slope).
The graph on the right, below, has
similar features — there is a constant, positive velocity
(as denoted by the constant, positive slope). However,
the slope of the graph on the right is larger than that
on the left and this larger slope is indicative of a larger
velocity.
The object represented by the graph on the
right is traveling faster than the object represented by
the graph on the left.
The principle of slope can be used
to extract relevant motion characteristics from a
position vs. time graph; as the slope goes, so goes the
velocity.
Slow,
Rightward (+)
Constant
Velocity
Fast,
Rightward (+)
Constant
Velocity
Example 2
Consider the graphs below as another
application of this
principle of slope.
The graph on the left, below, is
representative of an object which is moving with a
negative velocity (as denoted by the negative slope), a
constant velocity (as denoted by the constant slope), and
a small velocity (as denoted by the small slope).
The graph on the right, below, has similar features — there is a
constant, negative velocity (as denoted by the constant,
negative slope). However, the slope of the graph on the
right is larger than that on the left and once again, this
larger slope is indicative of a larger velocity.
The object represented by the graph on the right is traveling
faster than the object represented by the graph on the
left.
Slow, Leftward
(–)
Constant
Velocity
Fast, Leftward
(–)
Constant
Velocity
Example 3
As a final application of this principle of slope,
consider the two graphs below. Both graphs show plotted
points forming a curved line. Curved lines have changing
slope; they may start with a very small slope and begin
curving sharply (either upwards or downwards) towards a
large slope. In either case, the curved line of changing
slope is a sign of accelerated motion (i.e., changing
velocity).
Applying the principle of slope to the graph
on the left, below, you would conclude that the object depicted
by the graph is moving with a negative velocity (since
the slope is negative). Furthermore, the object starts
with a small velocity (the slope starts out
small) and finishes with a large velocity (the
slope becomes large). That means this object is
moving in the negative direction and speeding up (the
small velocity turns into a larger velocity). This is an
example of negative
acceleration – moving in the negative direction and
speeding up.
The graph on the right, below, also depicts an
object with negative velocity (since there is a negative
slope). The object begins with a large velocity (the slope
is initially large) and finishes with a small velocity
(the slope becomes smaller). This object is
moving in the negative direction and slowing down (the large velocity
turns into a smaller velocity). This
is an example of positive
acceleration – moving in a negative direction
and slowing down.
Leftward (–)
Velocity;
Slow to
Fast
Leftward (–)
Velocity;
Fast to
Slow
The principle of slope is an incredibly useful
principle for extracting relevant information about the
motion of objects as described by their position vs. time
graph. Once you've practiced the principle a few times,
it becomes a natural means of analyzing
position-time graphs.
Use the principle of slope to
describe the motion of the objects depicted by the two
plots below. In your description, be sure to include such
information as the direction of the velocity vector
(i.e., positive or negative), whether there is a constant
velocity or an acceleration, and whether the object is
moving slow, fast, from slow to fast or from fast to
slow. Be complete in your description.
