Science Vault - Year 11 HSC Physics

8.4 - Moving About

8.4.1 - Graphing Motion

displacement-time graphs / velocity-time graphs

Displacement-Time Graphs

The study of one-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. Here we will focus on the use of displacement 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 displacement vs. time graph. The first part of this section involves the study of the relationship between the motion of an object and the shape of its s-t graph.

To begin, consider a car moving with a constant, rightward (+) velocity of 10 ms-1.


If the position-time data for such a car were graphed, the resulting graph would look like the graph below. Note that a motion with constant, positive velocity results in a line of constant and positive slope when plotted as a displacement-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 displacement-time data for such a car were graphed, the resulting graph would look like the graph below. Note that a motion with changing, positive velocity results in a line of changing and positive slope when plotted as a displacement-time graph.

The displacement vs. time graphs for the two types of motion – constant velocity and changing velocity (acceleration) – are depicted as follows:

Positive Velocity and Constant Velocity

Positive Velocity and Changing Velocity
(acceleration)
 

The Principle of Slope for a s–t Graph

The shapes of the displacement 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 displacement-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 displacement vs. time graph; as the slope goes, so goes the velocity.

Slow, Rightward (+) Constant Velocity

Slow Fast, Rightward (+) Constant

Example 2

Consider the graphs below as another application of this principle of slope.

Slow, Leftward (–) Constant Velocity

Fast, Leftward (–) Constant Velocity

The graph on the left, above, 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 travelling faster than the object represented by the graph on the left.

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 displacement vs. time graph. Once you've practiced the principle a few times, it becomes a natural means of analysing position-time graphs.

Determining the Slope on a s-t Graph

You learned earlier that the slope of the line on a displacement vs. time graph is equal to the velocity of the object. If the object is moving with a velocity of +4 ms-1, then the slope of the line will be +4 ms-1. If the object is moving with a velocity of -8 ms-1, then the slope of the line will be -8 ms-1. If the object has a velocity of 0 ms-1, then the slope of the line will be 0 ms-1.

The importance of slope to a displacement vs. time graph means that a student of physics must have a good understanding of how to calculate the slope of a line. This section will discuss the method for determining the slope of a line on a displacement-time graph.

Let's begin by considering the displacement vs. time graph below.

The line slopes upwards to the right. But mathematically, by how much does it slope upwards along the vertical (position) axis per 1 second along the horizontal (time) axis? To answer this question use the slope equation:

The slope equation says that the slope of a line is found by dividing the amount of rise of the line between any two points by the amount of run of the line between the same two points. In other words,

  1. Pick two points on the line and determine their coordinates.
  2. Determine the difference in y-coordinates of these two points (rise).
  3. Determine the difference in x-coordinates of these two points (run).
  4. Divide the difference in y-coordinates (rise) by the difference in x-coordinates (run).
  5. Slope = rise/run.

The calculations below apply this method to determine the slope of the line in the graph above. Note that three different calculations are performed for three different sets of points on the line. In each case, the result is the same: the slope is 10 ms-1.

So that was easy — rise over run is all that is involved.

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Velocity-TIme Graphs

In this section we focus on the use of velocity 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 velocity vs. time graph. The first part of this section involves a study of the relationship between the motion of an object and the shape of its v-t graph.

Consider a car moving with a constant, rightward (+) velocity of +10 ms-1. As you learned in the previous section, a car moving with a constant velocity is a car moving with zero acceleration.

If the velocity-time data for such a car were graphed, the resulting graph would look like the graph below. Note that a motion with constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity.

Now consider a car moving with a rightward (+), changing velocity – that is, a car that is moving rightward and speeding up or accelerating. Since the car is moving in the positive direction and speeding up, it is said to have a positive acceleration.

If the velocity-time data for such a car were graphed, the resulting graph would look like the graph below. Note that a motion with changing, positive velocity results in a diagonal line when plotted as a velocity-time graph. The slope of this line is positive, corresponding to the positive acceleration. In addition, only positive velocity values are plotted, corresponding to a motion with positive velocity.

The velocity vs. time graphs for the two types of motion – constant velocity and changing velocity (acceleration) – can be summarized as follows:

Positive Velocity Zero Acceleration

Positive Velocity Positive Acceleration

The Principle of Slope for a v-t Graph

The shapes of the velocity 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 velocity-time graph reveals useful information about the acceleration of the object. Whatever characteristics the acceleration has, the slope will exhibit the same (and vice versa). If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). This principle can be extended to any motion conceivable.

The slope of a velocity-time graph reveals information about the object's acceleration. But how can you tell if the object is moving in the positive direction (i.e., positive velocity) or in the negative direction (i.e., negative velocity)? And how can you tell if the object is speeding up or slowing down? The answers to these questions hinge on your ability to read a graph.

Positive Velocity vs. Negative Velocity

Since the graph is a velocity-time graph, the velocity is positive whenever the line lies in the positive region (positive y-values, i.e. above the x-axis) of the graph. Similarly, the velocity is negative whenever the line lies in the negative region (negative y-values, i.e. below the x-axis) of the graph. As you learned previously, a positive velocity means the object is moving in the positive direction; and a negative velocity means the object is moving in the negative direction. So if an object is moving in the positive direction, the line is located in the positive region of the velocity-time graph (regardless if it is sloping up or sloping down). Likewise, an object is moving in the negative direction if the line is located in the negative region of the velocity-time graph (regardless if it is sloping up or sloping down). Finally, if a line crosses the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

Acceleration vs. Deceleration

How can you tell if the object is speeding up (acceleration) or slowing down (deceleration)? Speeding up means that the magnitude (the value) of the velocity is increasing. For instance, an object with a velocity changing from +3 ms-1 to + 9 ms-1 is speeding up. Similarly, an object with a velocity changing from -3 ms-1 to  -9 ms-1 is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign or direction) is increasing; the speed is getting larger.

Given this fact, an object is speeding up if the line on a velocity-time graph is changing from a location near the 0-velocity point to a location further away from the 0-velocity point. That is, if the line is moving away from the x-axis (the 0-velocity point), then the object is speeding up. Conversely, if the line is moving towards the x-axis, the object is slowing down.

The Meaning of Slope for a v-t Graph

As discussed in the previous section, the shape of a velocity vs. time graph reveals pertinent information about an object's acceleration. For example, if the acceleration is zero, then the velocity-time graph is a horizontal line (i.e., the slope is zero). If the acceleration is positive, then the line is an upward sloping line (i.e., the slope is positive). If the acceleration is negative, then the velocity-time graph is a downward sloping line (i.e., the slope is negative). If the acceleration is large, then the line slopes up steeply (i.e., the slope is large). Thus, the shape of the line on the graph (horizontal, sloped, steeply sloped, mildly sloped, etc.) is descriptive of the object's motion. The actual value of the slope of any straight line on a velocity-time graph corresponds to the acceleration of the object.

Relating Shape to Motion

This section will examine how the above principle applies to a variety of motions. In each diagram below, a short verbal description of a motion is given (e.g., "constant, rightward velocity") and an accompanying ticker tape diagram is shown. Next, the corresponding velocity-time graph is sketched and an explanation is given.

 

 

 

 

Determining the Slope on a v-t Graph

In this section we will discuss the method for calculating the slope of a line on a velocity-time graph. Let's begin by considering the velocity vs. time graph below.

The line slopes upwards to the right. But mathematically, by how much does it slope upwards along the vertical (velocity) axis per 1 second along the horizontal (time) axis? To answer this question use the slope equation:

The slope equation says that the slope of a line is found by dividing the amount of rise of the line between any two points by the amount of run of the line between the same two points. In other words:

  1. Pick two points on the line and determine their coordinates.
  2. Determine the difference in y-coordinates of these two points (rise).
  3. Determine the difference in x-coordinates of these two points (run).
  4. Divide the difference in y-coordinates (rise) by the difference in x-coordinates (run).
  5. Slope = rise/run.

The calculation below applies this method to determine the slope of the line in the graph above. Note that three different calculations are performed for three different sets of points on the line. In each case,

Determining the Area on a v-t Graph

As you learned in an earlier section of this lesson, a plot of velocity vs. time can be used to determine the acceleration of an object (slope = acceleration). In this section, you will learn how a plot of velocity vs. time can also be used to determine the distance travelled by an object. For velocity vs. time graphs, the area bounded by the line and the axes represents the distance travelled.

The diagram below shows three different velocity-time graphs; the shaded regions between the line and the axes represent the distance travelled during the stated time interval.

The shaded area is representative of the distance travelled by the object during the time interval from 0 seconds to 6 seconds. This representation of the distance travelled takes on the shape of a rectangle whose area can be calculated using the appropriate equation.

 

The shaded area is representative of the distance travelled by the object during the time interval from 0 seconds to 4 seconds. This representation of the distance travelled takes on the shape of a triangle whose area can be calculated using the appropriate equation.

 

The shaded area is representative of the distance travelled by the object during the time interval from 2 seconds to 5 seconds. This representation of the distance travelled takes on the shape of a trapezoid whose area can be calculated using the appropriate equation.

 

The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle or a trapezoid. Area formulae for each shape are given below.

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download worksheet 8.4.1.B - displacement-time graphs

download worksheet 8.4.1.C - velocity-time graphs

download experiment 2 - recording and graphing motion

download video for experiment 2

 
References

Internet References

  • Under constructtion.

Syllabus References

These references relate to the content covered on this page and can be found in Section 8.4.1 of the syllabus.

1. Vehicles do not typically tavel at a constant speed.

Students learn to:

  • Present information graphically of:

    – displacement vs time
    – velocity vs time

    for objects with uniform and non-uniform linear velocity

Textbook References

Taken from:

Heffernan, D., Parker, A., Pinniger, G. & Harding, J. (2002) Physics Contexts 1, Pearson Education, Melbourne

  • Sections x and x on pp. x - x