Speed and Velocity, Motion in One Dimension | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:
- (B) describe and analyze motion in one dimension using equations with the concepts of distance, displacement, speed, average velocity, instantaneous velocity, and acceleration.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Position and Speed of an Object, as well as the following standards:

(4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The student is expected to:
- (B) describe and analyze motion in one dimension using equations with the concepts of distance, displacement, speed, average velocity, instantaneous velocity, and acceleration.

Section Key Terms

average speed	average velocity	instantaneous speed
instantaneous velocity	speed	velocity

Teacher Support

In this section, students will apply what they have learned about distance and displacement to the concepts of speed and velocity.

[BL][OL] Before students read the section, ask them to give examples of ways they have heard the word speed used. Then ask them if they have heard the word velocity used. Explain that these words are often used interchangeably in everyday life, but their scientific definitions are different. Tell students that they will learn about these differences as they read the section.

[AL] Explain to students that velocity, like displacement, is a vector quantity. Ask them to speculate about ways that speed is different from velocity. After they share their ideas, follow up with questions that deepen their thought process, such as: Why do you think that? What is an example? How might apply these terms to motion that you see every day?

Speed

There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we will look at time, speed, and velocity to expand our understanding of motion.

A description of how fast or slow an object moves is its speed. Speed is the rate at which an object changes its location. Like distance, speed is a scalar because it has a magnitude but not a direction. Because speed is a rate, it depends on the time interval of motion. You can calculate the elapsed time or the change in time, $Δ t$ , of motion as the difference between the ending time and the beginning time

Δ t = t_{f} - t_{0} .

The SI unit of time is the second (s), and the SI unit of speed is meters per second (m/s), but sometimes kilometers per hour (km/h), miles per hour (mph) or other units of speed are used.

When you describe an object's speed, you often describe the average over a time period. Average speed, v_avg, is the distance traveled divided by the time during which the motion occurs.

v_{avg} = \frac{distance}{time}

You can, of course, rearrange the equation to solve for either distance or time

time = \frac{distance}{v_{avg}} .

distance = v_{avg} × time

Suppose, for example, a car travels 150 kilometers in 3.2 hours. Its average speed for the trip is

\begin{matrix} v_{avg} & = & \frac{distance}{time} \\ = & \frac{150 km}{3.2 h} \\ = & 47 km/h. \end{matrix}

A car's speed would likely increase and decrease many times over a 3.2 hour trip. Its speed at a specific instant in time, however, is its instantaneous speed. A car's speedometer describes its instantaneous speed.

View original figure (external; image not reproduced in this text edition)

Original figure description: A drawing is shown of a house on the left and a store on the right. The distance between the two is labeled three kilometers. A double-arrow vector between the house and the store is labeled with the equation change in d total equals zero.

During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, because there was no net change in position.

Practice Problems

A pitcher throws a baseball from the pitchers mound to home plate in 0.46 s. The distance is 18.4 m. What was the average speed of the baseball?

40 m/s
- 40 m/s
0.03 m/s
8.5 m/s

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Velocity

The vector version of speed is velocity. Velocity describes the speed and direction of an object. As with speed, it is useful to describe either the average velocity over a time period or the velocity at a specific moment. Average velocity is displacement divided by the time over which the displacement occurs.

v_{avg} = \frac{displacement}{time} = \frac{Δ d}{Δ t} = \frac{d_{f} - d_{0}}{t_{f} - t_{0}}

Velocity, like speed, has SI units of meters per second (m/s), but because it is a vector, you must also include a direction. Furthermore, the variable v for velocity is bold because it is a vector, which is in contrast to the variable v for speed which is italicized because it is a scalar quantity.

Suppose a passenger moved toward the back of a plane with an average velocity of –4 m/s. We cannot tell from the average velocity whether the passenger stopped momentarily or backed up before he got to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals such as those shown in Figure_02_02_plane. If you consider infinitesimally small intervals, you can define instantaneous velocity, which is the velocity at a specific instant in time. Instantaneous velocity and average velocity are the same if the velocity is constant.

View original figure (external; image not reproduced in this text edition)

Original figure description: A drawing of an airplane facing to the right is shown, and is set in quadrant one of a coordinate plane. The x-axis is labeled df and then do. The distance between df and do is labeled with a left-pointing vector, change in d total. Four other vectors are shown below the airplane. Vector change in d a points to the left and is short. Vector change in d b points to the left and is also short. Vector change in d c points to the right and is short. Vector change in d d points to the left and is longer.

The diagram shows a more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip.

Earlier, you have read that distance traveled can be different than the magnitude of displacement. In the same way, speed can be different than the magnitude of velocity. For example, you drive to a store and return home in half an hour. If your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero because your displacement for the round trip is zero.

Calculating Average Velocity

A student has a displacement of 304 m north in 180 s. What was the student's average velocity?

Solution

v_{avg} = \frac{Δ d}{Δ t} = \frac{304 m}{180 s} = 1.7 m/s north

Discussion

Since average velocity is a vector quantity, you must include direction as well as magnitude in the answer. Notice, however, that the direction can be omitted until the end to avoid cluttering the problem. Pay attention to the significant figures in the problem. The distance 304 m has three significant figures, but the time interval 180 s has only two, so the quotient should have only two significant figures.

Solving for Displacement when Average Velocity and Time are Known

Layla jogs with an average velocity of 2.4 m/s east. What is her displacement after 46 seconds?

Solution

\begin{matrix} v_{avg} & = & \frac{Δ d}{Δ t} \\ Δ d & = & v_{a v g} Δ t \\ = & (2.4 m/s)(46 s) \\ = & 1.1 \times 10^{2} m east \end{matrix}

Discussion

The answer is about 110 m east, which is a reasonable displacement for slightly less than a minute of jogging. A calculator shows the answer as 110.4 m. We chose to write the answer using scientific notation because we wanted to make it clear that we only used two significant figures.

Solving for Time when Displacement and Average Velocity are Known

Phillip walks along a straight path from his house to his school. How long will it take him to get to school if he walks 428 m west with an average velocity of 1.7 m/s west?

Solution

\begin{matrix} v_{avg} & = & \frac{Δ d}{Δ t} \\ Δ t & = & \frac{Δ d}{v_{avg}} \\ = & \frac{428 m}{1.7 m/s} \\ = & 2.5 \times 10^{2} s \end{matrix}

Discussion

Here again we had to use scientific notation because the answer could only have two significant figures. Since time is a scalar, the answer includes only a magnitude and not a direction.

Practice Problems

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A bird flies with an average velocity of 7.5 m/s east from one branch to another in 2.4 s. It then pauses before flying with an average velocity of 6.8 m/s east for 3.5 s to another branch. What is the birds total displacement from its starting point?

42 m west
6 m west
6 m east
42 m east

Check Your Understanding

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If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity, and under what circumstances are these two quantities the same?

Average speed. Both are the same when the car is traveling at a constant speed and changing direction.
Average speed. Both are the same when the speed is constant and the car does not change its direction.
Magnitude of average velocity. Both are same when the car is traveling at a constant speed.
Magnitude of average velocity. Both are same when the car does not change its direction.

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Section Summary

Average speed is a scalar quantity that describes distance traveled divided by the time during which the motion occurs.
Velocity is a vector quantity that describes the speed and direction of an object.
Average velocity is displacement over the time period during which the displacement occurs. If the velocity is constant, then average velocity and instantaneous velocity are the same.

Key Equations

Average speed	$v_{avg} = \frac{distance}{time}$
Average velocity	$v_{avg} = \frac{Δ d}{Δ t} = \frac{d_{f} - d_{0}}{t_{f} - t_{0}}$

Concept Items

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Problems

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Critical Thinking

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Terri, Aaron, and Jamal all walked along straight paths. Terri walked 3.95 km north in 48 min. Aaron walked 2.65 km west in 31 min. Jamal walked 6.50 km south in 81 min. Which of the following correctly ranks the three boys in order from lowest to highest average speed?

Jamal, Terri, Aaron
Jamal, Aaron, Terri
Terri, Jamal, Aaron
Aaron, Terri, Jamal

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Test Prep Multiple Choice

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A car travels with an average speed of 23 m/s for 82 s. Which of the following could NOT have been the car's displacement?

1,700 m east
2,000 m west
1,600 m north
1,500 m south

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Test Prep Short Answer

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What can you infer from the statement, Velocity of an object is zero?

Object is in linear motion with constant velocity.
Object is moving at a constant speed.
Object is either at rest or it returns to the initial point.
Object is moving in a straight line without changing its direction.

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Test Prep Extended Response

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A father and his daughter go to the bus stop that is located 75 m from their front door. The father walks in a straight line while his daughter runs along a varied path. Despite the different paths, they both end up at the bus stop at the same time. The fathers average speed is 2.2 m/s, and his daughters average speed is 3.5 m/s. (a) How long does it take the father and daughter to reach the bus stop? (b) What was the daughters total distance traveled? (c) If the daughter maintained her same average speed and traveled in a straight line like her father, how far beyond the bus stop would she have traveled?

(a) 21.43 s (b) 75 m (c) 0 m
(a) 21.43 s (b) 119 m (c) 44 m
(a) 34 s (b) 75 m (c) 0 m
(a) 34 s (b) 119 m (c) 44 m

Glossary

average speed: distance traveled divided by time during which motion occurs
average velocity: displacement divided by time over which displacement occurs
instantaneous speed: speed at a specific instant in time
instantaneous velocity: velocity at a specific instant in time
speed: rate at which an object changes its location
velocity: the speed and direction of an object

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