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# 9.1: Linear Motion and How to Describe It

Difficulty Level: At Grade Created by: CK-12

This section will teach you how to characterize one-dimensional motion by appreciating the use and construction of its representation using graphs.

Understanding how things move is fundamental to our understanding of the physical universe. Critical to this understanding is the ability to portray motion in a manner that is clear, accurate, precise, efficient, and reproducible. “Linear Motion and How to Describe It” identifies the terms used to characterize motion and illustrates the graphical methods used to represent motion visually. The chapter is concerned with the “kinematics” of motion, without regard to the cause of the motion (i.e., no mention of forces).

## Learning Objectives

• to clarify the terms you use to characterize motion and to show their relationships
• to connect your physical ideas of motion to a graphical representation of motion
• to boost your ability to graph motion through tutorial exercises using a motion sensor

## Introduction to One-Dimensional Motion

You are familiar with motion—a dog chasing a cat, a truck backing up, an apple falling from a tree, people walking in a park—these are just a few examples of motion of a mass or body. In this chapter you are going to study motion, a word synonymous with movement—the state of a body (an animate or inanimate mass) when not at rest. Of course, the straight-line (or “linear”) motions mentioned here are not the only types of motions that can occur. Rotational motion (such as the spinning of a bicycle wheel) and vibrational motion (such as the oscillations of a butterfly wing), or any combinations of linear, rotational, and vibrational motion are also possible. In this chapter, however, we are interested only in linear motion because it is the simplest type of motion and it provides a framework upon which more complicated types of motion of bodies can be characterized.

## Verbal Description of Linear Motion

So how can you describe motion? Well, you can use verbal and/or written descriptions. Common terms used such as “speed,” “velocity,” “acceleration,” and “deceleration” come to mind, which are used, sometimes interchangeably, to describe the motion of a body. “Direction" is also a property of linear motion and can be dealt with simply by using algebraic signs, "$+$" for one direction (e.g., East), "$-$" for the opposite one (e.g., West) (analogous to a Cartesian coordinate system). Consider the following example:

Example 1 An oil truck, sitting at rest at time $t = 0$ seconds (s) is traveling at a speed of $3$ meters per second $(3 \;\mathrm{m/s}), 6 \;\mathrm{s}$ later.

Question Is the description of the motion of the oil truck in Example 1 the most complete description of its motion? For one thing, you do not know how the oil truck got from rest to a speed of $3 \;\mathrm{m/s}$. What type of motion occurred during that $6s$ time period? In which direction was the oil truck traveling? What reference points were being used to locate the truck’s position at every point in its motion? What was the speed of the truck at a particular instant of time in the $6s$ period?

There is a saying “one picture is worth a thousand words.” Hence, is there a less ambiguous way, other than using just words, to describe the motion of the oil truck in the 6s? The answer is yes and that representation is a visual one. However, before we introduce visual representations of the motion, we need to standardize our definitions of the terms used to describe motion.

## Vocabulary

distance, average speed, position, velocity, acceleration

Distance
Distance is the amount of length that a body has moved from one instant of time to another instant of time. It is always a positive number because it is just an amount, and says nothing about a direction. For example: $30\;\mathrm{m}, 35.6\;\mathrm{ft}, 25.138\;\mathrm{cm}$.
Average Speed
Speed, in general, is a measure of how fast a body is moving without regard to the direction of motion. Average speed (symbol, $s_{ave}$) is defined as the total distance a body travels per unit time interval. Because distance and time, $t$, are positive quantities, speed is always a positive quantity. We can express $s_{ave}$ mathematically in the following way:

$s_{ave} = \frac{\text{Total distance traveled}}{\text{Total time elapsed}}$

Example 2
The car travels an average of $42$ miles in one hour:

$w_{ave}=42 \frac{\text{mi}}{\text{h}}$

Example 3
The bird flies $50.6$ feet in $10.0$ seconds on average:

$s_{ave}=5.06 \ \frac{\text{ft}}{\text{s}}$

Example 4
On average, a bug runs $25.276$ centimeters in $2$ minutes:

$s_{ave}=12.6 \ \frac{\text{cm}}{\text{min}}$

Instantaneous speed (symbol $s$) is a body’s speed at a particular point in time. An oil truck’s speedometer displays the truck’s instantaneous speed.
Position
Position refers to the location of a body at one instant of time with respect to some reference position. It is a vector—meaning it is both a magnitude and a direction. Vector quantities usually have symbols that are written in boldface type, which we will use here. A Cartesian coordinate system provides a convenient reference frame for you to use to locate a position. In this case, position can have either a positive or a negative value.
Example 5
$y=40 \;\mathrm{mi}$ directly north of home
Example 6
$x=-\!6.0 \;\mathrm{cm}$ along the $x-$axis in a Cartesian coordinate system
Example 7
$A= 39.7 \;\mathrm{m}$ at an angle of $25^\circ$ east of the $+\!y-$ axis in a Cartesian coordinate system
Velocity
Velocity is a term used to specify not only the speed of a body, but also its direction. Like position, velocity is also a vector. The Cartesian coordinate system provides a convenient reference frame for its direction. In this case, velocity can have either a positive or a negative value. There are two types of velocity, average velocity (symbol $v_{ave}$) and instantaneous velocity (symbol $v$").
Average velocity is defined as the change in the position (called a “displacement”) of a body during a particular time interval. Because position is a vector, average velocity can be positive or negative in a Cartesian coordinate system. The average velocity, written in terms of the change (symbol $\Delta$) in the initial position, $x_i$, and the final position, $x_f$, is:

$v_{ave} = \frac{\Delta x}{\Delta t} = \frac{x_f-x_i}{t_f-t_i}$

Example 8
$v_{ave}=40 \ \frac{\text{mi}}{\text{h}}\text{ directly north}$
Example 9
$v_{ave}=-0.20 \ \frac{\text{mm}}{\text{s}}$
Example 10
$v_{ave}=36.7 \ \frac{\text{cm}}{\text{s}}$ at an angle of $25^\circ$ above the $+\!x \!-$ axis

Instantaneous velocity (boldface symbol $v$) refers to the velocity of a body at one particular instant of time. If the direction is not specified (as in the oil truck’s “speedometer”), then Instantaneous Velocity and instantaneous speed are equivalent.

Example 11
$v=30\ \frac{m}{s} \text{ northward}$
Example 12
$v=20 \ \frac{mi}{h}$
Example 13
$v=6.0 \ \frac{m}{s} \text{ in the }+\!y- \text{direction}$
Acceleration
When there is a change in the instantaneous velocity of a body during a particular time interval, the body possesses an average acceleration (symbol $a_{ave}$). Because velocity is a vector, average acceleration can be positive or negative in a Cartesian coordinate system. The average acceleration, written in terms of the change (symbol $\Delta$) in the initial velocity, $v_i$, and the final velocity, $v_f$, is:

$a_{ave}=\frac{\Delta v}{\Delta t} = \frac{v_f-v_i}{t_f-t_i}$

The instantaneous acceleration (symbol $a$) is the acceleration of a body a particular instant of time. In this lesson we will only consider accelerations that are constant in time. For that reason, $a = a_{ave}$.

Example 14
$a=30 \ \frac{m}{s^2} \text{ northward}$
Example 15
$a=20 \ \frac {mi}{h^2}$
Example 16
$a=6.0 \ \frac{m}{s^2} \ \text{in the} +\!y \text{-direction}$

## Visual Descriptions of One-Dimensional Motion

Sometimes you cannot use words alone to accurately describe the motion of a body. You need to convey the motion in a more visual manner. This can be done in two ways:

### 1. The Motion Diagram

In this visual description, a body’s one-dimensional motion is represented by a sequence of dots. The distance between each dot represents the body’s change in position during that time interval. The time interval is established by the device that creates the dots. Large distances between adjacent dots indicate that the body was moving fast during that time interval. Small distances between adjacent dots indicate that the body was moving slow during that time interval. A constant distance between dots indicates that the body is moving with constant velocity and not accelerating. A changing distance between dots indicates that the body is changing velocity and is thus accelerating.

Example 17

Suppose the oil truck mentioned above drips oil at a regular rate of time as the truck travels along its route.

Figure 1 shows two of many possible motion diagrams of the oil truck during the first 6 seconds of its motion. The truck is moving from left to right. There is a 1 second time interval between dots.

A motion diagram of the oil truck dripping oil at a constant time interval. In (a) the distance between adjacent dots increases successively by a factor of as time increases from left to right. This indicates that the oil truck is increasing its speed (accelerating) at each successive interval. In (b) the distance between adjacent dots are equal for the first of motion, indicating that the oil truck is moving with constant speed. For the last of motion the distance between adjacent dots are also equal but larger in length, indicating a greater constant speed.

You can make your own motion diagrams. An inexpensive “drip tube” (used for watering plants) filled with molasses or soy sauce can be used to approximate a constant drip rate device (sort of like a “water clock”). You can also cut a $20$ oz plastic soda bottle in half at its midsection, stick a cork tightly into the smaller mouth of the bottle, and then drill a hole into the cork, just large enough to insert a medicine dropper. You can then fill the empty portion of the bottle with the fluid and the medicine dropper will be your dripper.

The advantages of using motion diagrams are that you get a quick, visual idea of the type of motion involved. You can determine average speeds or average velocities, but not instantaneous speeds or instantaneous velocities. Also, for very long periods of motion, motion diagrams become impractical because of the quantity of dots involved and the time needed to analyze the dots.

### 2. Graphing Motion

Unlike motion diagrams, graphs provide more accurate information by providing a “continuous” visual description of motion. Graphing motion usually involves making a two-dimensional plot of an instantaneous variable (distance, position, velocity, or acceleration) as a function of time. Average values of these variables can also be determined from these graphs.

Let us now return to the oil truck, which started from rest and was eventually traveling at $3 \;\mathrm{m/s}$ in $6 \;\mathrm{s}$. Three possible ways in which this motion could be interpreted are as follows:

1. Starting from rest, the truck immediately traveled at a constant speed of $3 \;\mathrm{m/s}$ for the next $6 \;\mathrm{s}$ (virtually impossible to do).
2. Starting from rest, the truck steadily increased its speed, reaching a speed of $3 \;\mathrm{m/s}$ in $6 \;\mathrm{s}$.
3. The truck’s instantaneous speed is $3 \;\mathrm{m/s}$ at $t = 6 \;\mathrm{s}$ as indicated by the truck’s speedometer (or by police radar). The truck’s speed could have been any value before $6 \;\mathrm{s}$.

Which interpretation is the correct one? We can answer this question if we have a graphical description of the motion for each of these three possible interpretations.

Interpretation 1

Let’s start by looking at a distance versus time (and/or position versus time) graph for the oil truck based on interpretation 1. Figure 2(a) shows the oil truck’s distance increasing at a constant rate as a function of time, starting from rest. The rate of speed is determined by the slope of the red line, which is positive.

Notice that we can determine the exact distance, $D$, that the oil truck has moved at each instant of time in a continuous manner. With the motion diagrams, we could only determine the distances the oil truck moves at discrete instances of time.

The oil truck could also be moving in the opposite direction. In this case, we could plot a position versus time graph that would show the oil truck moving in the negative x direction, another possible motion based on interpretation 1. Figure 2(b) shows the oil truck moving in the negative x direction at a constant rate of speed. It is also moving with a constant negative velocity based on the slope of the red line.

A Distance Versus Time Graph

Interpretation 2

Now let us turn our attention to interpretation $2$ of the oil truck’s motion in the 6 s period. Figure 3(a) shows a distance versus time graph (in red) in which the truck’s distance increases at a greater rate as time increases from $t = 0$ to $t = 6 \;\mathrm{s}$. Therefore, the speed of the truck also increases, but at a constant rate, as shown by the increasing “slope” of the tangent lines (small black lines) to the (red) curve. The instantaneous speed $(3 \;\mathrm{m/s})$ at $t = 6 \;\mathrm{s}$ would be the slope of the tangent line to the (red) curve right at $t = 6 \;\mathrm{s}$.

Figure 3(b) shows the same distance versus time graph of the motion for the oil truck as in (a). The average speed, $v_{ave}$, between any two points on a distance versus time curve can be obtained by determining the slope of the line connecting those two points (in black).

Graphs of distance, , in meters versus time, , in seconds for the oil truck. In (a) the truck’s distance increases at a greater rate as time increases from to . The speed of the truck also increases, but at a constant rate, as shown by the increasing “slope” of the tangent lines (the small black lines) to the red curve. In (b) the average speed between times and , determined by the slope of the black line, is .

Interpretation 3

In this interpretation, a distance versus time graph (or a position versus time graph) could show any shape as long as the slope of the tangent line to the curve at $t = 6 \;\mathrm{s}$ gives a value of $3 \;\mathrm{m/s}$. Rather than determining tangents to the curve in these graphs at various points in the motion, it would be better to plot the speed (or velocity) versus time of the oil truck.

Figure 4 (a) shows a velocity versus time graph (in red) of the motion of the oil truck. In this graph, the oil truck first accelerates at a constant rate $a_{ave} = 0.67\;\mathrm{m/s}^2$, then accelerates at a constant rate of $a_{ave} = 0.33\;\mathrm{m/s}^2$ to a final velocity of $3 \;\mathrm{m/s}$.

Figure 4 (b) is an acceleration versus time graph of the motion of the oil truck based on the information in Figure 4(a).

From these few examples, we can see now how graphing can be used to give us a more complete description of the motion of a body.

Graphs of the velocity of the oil truck versus time . In Figure 4(a), the truck’s velocity is zero at time . As time increases, the truck’s velocity increases at a constant rate until a velocity is reached at . After that time, the truck’s velocity increases at a lower rate until it reaches a velocity at . Graph (b) shows that the truck’s average acceleration from to , and an average acceleration , from to .

## Graphing of Motion—A Tutorial Exercise

This tutorial exercise is designed to advance your ability to graph motion. A motion sensor will be used to detect the motion of a body, in this case YOU, and that motion will be graphed by a computer.

Equipment

• Motion sensor with computer interface box and cables
• Desktop or laptop computer
• Table or support stand

Experimental Setup

The experimental setup used to graph your motion is shown in Figure 5. A motion sensor is connected to an interface box which in turn is connected to a computer. The interface box translates the signals from the motion sensor into the computer. The computer displays these signals, either as a position, a velocity, or acceleration as a function of time.

Experimental setup for motion sensing. A motion sensor, interfaced to a computer, is directed at the midsection of a student. The student moves toward or away from the motion sensor and the sensor monitors the student’s movement. A computer gives a graphical display of the motion.

How the Motion Sensor Works

When describing the motion of an object, knowing where the object is relative to a reference point, how fast and in what direction it is moving, and how it is accelerating (changing its rate of motion) is crucial. The motion sensor is a sonar ranging device using high-frequency pulses of sound that reflect from an object to determine the position of the object. The ultrasound pulses travel at a constant speed ($\sim 343 \;\mathrm{m/s}$ in air at room temperature). As the object moves, the change in its position is measured many times each second as the pulse travels back and forth from object to sensor.

Positioning the Motion Sensor and Computer

Mount the motion sensor on a table or support rod so that it is aimed at your midsection when you are standing in front of the sensor. Clear the area for at least $3$ meters (about $9$ feet) in front of the motion sensor. Position the computer monitor so you or your lab partner can see the screen while you move in front of the motion sensor.

General Procedure

In this activity, the motion sensor will measure your position, velocity, or acceleration as you move. The computer plots your position, $x$, on a graph as a function of time, $t$.

Moving away from the motion sensor could be considered motion in the positive $x-$direction, and moving toward the sensor considered motion in the negative $x-$direction.

Tips for Better Data Acquisition

• Always stay in line directly in front of the motion sensor when at rest or when in motion. Try to avoid unnecessary movements that might be sensed.
• Be sure that the area around you is clear of all obstacles that may interfere with the motion sensor and cause a false reading.
• Never stand closer than $0.5 \;\mathrm{m}$ or farther than $4.0 \;\mathrm{m}$ from the motion sensor. Otherwise, your position will not be correctly determined by the motion sensor.
• Starting at $0.5 \;\mathrm{m}$ in front of the motion sensor (your $x = 0$ position) use masking tape to mark the floor at $0.5 \;\mathrm{m}$ intervals going away from the motion sensor for a total of $3.0 \;\mathrm{m}$.
• Once you have marked positions on the floor and you want the detector to produce readings that agree, stand at the $2.0 \;\mathrm{m}$ mark on the number line and have someone reposition the motion sensor until the reading on the computer shows a position $x = 2.0 \;\mathrm{m}$.
• Complete your drawings on the graphs in an idealized form rather than showing many small wiggles.
Note: It is very difficult to obtain accurate acceleration versus time graphs with the current motion sensors available due to the nature of the sensor.

Procedural Steps

1. Figure 6 shows six columns: (a) through $(f)$. Each column is headed by a “Description of motion” of your motion or a set of empty lines.
2. Below each description of motion are three graphs: a position versus time graph, a velocity versus time graph, and an acceleration versus time graph. They represent your motion in front of the motion sensor. Some graphs are complete, others are to be completed.
3. The challenge of these tutorial exercises is to predict the descriptions of the motion, to complete the remaining graphs based on the information given, and to write a description of the motion in the empty lines at the head of particular columns. Complete each column with your predictions one at a time, instead of checking several problems at once. Use the motion sensor to check your answers. Figure 7 shows the correct answers.

Description of Motion: graphs a and b

Description of Motion: graphs c and d

Description of Motion: graphs e and f

## Review Questions

1. In a position versus time graph, the data shows linear behavior that is negatively sloped with respect to the time axis. Which kind of motion is being represented by this data? (Circle one)
1.Constant speed and constant acceleration 2.Zero speed and constant acceleration 3.Increasing speed and constant acceleration 4.Constant speed and zero acceleration 5.Increasing speed and increasing acceleration
1.Can a body be slowing down while acceleration is increasing in magnitude? 2.Can a body be speeding up while its acceleration is decreasing in magnitude? 3.Can a body have non-zero instantaneous acceleration and zero velocity? 4.Can a body have zero acceleration and non-zero velocity?

Feb 23, 2012

Nov 25, 2014