The Reading on a Speedometer Represents the Average Velocity of the Car.

Department Learning Objectives

By the cease of this section, you will exist able to do the post-obit:

• Summate the boilerplate speed of an object
• Relate displacement and average velocity

Teacher Support

Instructor Support

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

• (4) Science concepts. The student knows and applies the laws governing motion in a variety of situations. The educatee is expected to:
  • (B) draw and analyze motility in ane dimension using equations with the concepts of altitude, displacement, speed, average velocity, instantaneous velocity, and dispatch.

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

• (4) Science concepts. The educatee knows and applies the laws governing motion in a variety of situations. The pupil is expected to:
  • (B) depict and analyze motion in ane 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

Teacher Support

In this section, students will apply what they accept learned almost distance and deportation to the concepts of speed and velocity.

[BL] [OL] Earlier students read the department, ask them to requite examples of ways they have heard the give-and-take speed used. And 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 deportation, is a vector quantity. Ask them to speculate near ways that speed is dissimilar from velocity. After they share their ideas, follow upward with questions that deepen their thought process, such as: Why practise y'all call back that? What is an example? How might apply these terms to motion that yous see every day?

Speed

There is more to move 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 fourth dimension, speed, and velocity to aggrandize our agreement 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 considering it has a magnitude but not a direction. Because speed is a rate, it depends on the time interval of motility. You can summate the elapsed time or the change in time, $\Delta t$ , of move as the difference between the catastrophe time and the first time

$$\Delta t=t_{\text{f}}-t_{\text{0}}.$$

The SI unit of fourth dimension is the 2nd (s), and the SI unit of speed is meters per second (m/s), just sometimes kilometers per hour (km/h), miles per hour (mph) or other units of speed are used.

When you describe an object'south speed, you frequently describe the boilerplate over a time period. Boilerplate speed, 5 _avg , is the distance traveled divided by the time during which the motion occurs.

$$5_{\text{avg}}=\frac{\text{altitude}}{\text{time}}$$

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

$$\text{time =}\frac{\text{distance}}{v_{\text{avg}}}\text{.}$$

$$\text{altitude =}{v}_{\text{avg}}\text{ × time}$$

Suppose, for case, a motorcar travels 150 kilometers in 3.ii hours. Its average speed for the trip is

$$\begin{array}{ccc}\hfill v_{\text{avg}}& =& \frac{\text{distance}}{\text{fourth dimension}}\hfill \\ & =& \frac{150\text{ km}}{\mathrm{iii.two}\text{ h}}\hfill \\ & =& 47\text{ km/h.}\hfill \end{array}$$

A automobile's speed would likely increase and decrease many times over a three.2 hr trip. Its speed at a specific instant in time, however, is its instantaneous speed. A car'due south speedometer describes its instantaneous speed.

Teacher Support

Instructor Back up

[OL] [AL] Caution students that average speed is not e'er the average of an object'southward initial and final speeds. For example, suppose a car travels a distance of 100 km. The first 50 km it travels 30 km/h and the second 50 km it travels at 60 km/h. Its average speed would be distance /(fourth dimension interval) = (100 km)/[(50 km)/(30 km/h) + (50 km)/(threescore km/h)] = forty km/h. If the auto had spent equal times at 30 km and 60 km rather than equal distances at these speeds, its boilerplate speed would have been 45 km/h.

[BL] [OL] Circumspection students that the terms speed, average speed, and instantaneous speed are all ofttimes referred to only every bit speed in everyday linguistic communication. Emphasize the importance in science to use correct terminology to avert confusion and to properly communicate ideas.

Figure 2.8 During a xxx-minute round trip to the store, the total altitude traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, because there was no cyberspace modify in position.

Worked Example

Calculating Average Speed

A marble rolls v.ii 1000 in 1.8 southward. What was the marble's average speed?

Strategy

We know the distance the marble travels, v.ii thousand, and the time interval, ane.8 s. We tin use these values in the average speed equation.

Give-and-take

Average speed is a scalar, so nosotros do not include direction in the answer. We can check the reasonableness of the answer by estimating: 5 meters divided by ii seconds is two.v one thousand/s. Since 2.5 m/south is close to 2.9 k/southward, the answer is reasonable. This is about the speed of a brisk walk, so information technology likewise makes sense.

Practice Problems

9 .

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

1. xl chiliad/due south
2. - 40 m/s
3. 0.03 m/s
4. 8.5 m/s

10 .

Cassie walked to her friend's firm with an average speed of 1.xl yard/due south. The distance between the houses is 205 k. How long did the trip take her?

1. 146 s

2. 0.01 s

3. 2.50 min

4. 287 s

Velocity

The vector version of speed is velocity. Velocity describes the speed and management of an object. Equally with speed, it is useful to describe either the boilerplate velocity over a fourth dimension period or the velocity at a specific moment. Boilerplate velocity is displacement divided past the time over which the displacement occurs.

$$v_{\text{avg}}=\frac{\text{displacement}}{\text{time}}=\frac{\Delta d}{\Delta t}=\frac{d_{\text{f}}-d_0}{t_{\text{f}}-t_0}$$

Velocity, similar speed, has SI units of meters per second (k/southward), merely considering it is a vector, you must likewise include a direction. Furthermore, the variable v for velocity is assuming because it is a vector, which is in contrast to the variable v for speed which is italicized because it is a scalar quantity.

Tips For Success

Information technology is important to keep in mind that the average speed is not the same matter as the average velocity without its direction. Similar we saw with displacement and distance in the concluding section, changes in direction over a time interval have a bigger effect on speed and velocity.

Suppose a passenger moved toward the back of a plane with an average velocity of –iv chiliad/s. We cannot tell from the average velocity whether the rider stopped momentarily or backed up before he got to the dorsum of the aeroplane. To get more details, we must consider smaller segments of the trip over smaller time intervals such equally those shown in Effigy 2.9. If you consider infinitesimally small intervals, you can define instantaneous velocity, which is the velocity at a specific instant in time. Instantaneous velocity and boilerplate velocity are the same if the velocity is constant.

Figure 2.9 The diagram shows a more detailed tape of an airplane passenger heading toward the back of the aeroplane, showing smaller segments of his trip.

Before, yous have read that distance traveled tin can exist unlike 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 hr. If your car'southward odometer shows the full distance traveled was six km, and so your average speed was 12 km/h. Your average velocity, all the same, was zip considering your displacement for the circular trip is zero.

Watch Physics

Calculating Average Velocity or Speed

This video reviews vectors and scalars and describes how to calculate boilerplate velocity and average speed when y'all know displacement and change in time. The video likewise reviews how to catechumen km/h to yard/s.

Which of the following fully describes a vector and a scalar quantity and correctly provides an example of each?

1. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Displacement is an example of a scalar quantity and fourth dimension is an case of a vector quantity.

2. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Time is an example of a scalar quantity and displacement is an case of a vector quantity.

3. A scalar quantity is fully described past its magnitude and management, while a vector needs only magnitude to fully describe information technology. Displacement is an example of a scalar quantity and time is an example of a vector quantity.

4. A scalar quantity is fully described by its magnitude and direction, while a vector needs but magnitude to fully describe information technology. Time is an case of a scalar quantity and displacement is an example of a vector quantity.

Instructor Support

Teacher Support

This video does a good task of reinforcing the difference between vectors and scalars. The educatee is introduced to the idea of using 'southward' to denote displacement, which yous may or may not wish to encourage. Before students watch the video, bespeak out that the instructor uses $\overrightarrow{s}$ for displacement instead of d, as used in this text. Explain the use of pocket-size arrows over variables is a common mode to denote vectors in higher-level physics courses. Caution students that the customary abbreviations for hour and seconds are not used in this video. Remind students that in their ain work they should employ the abbreviations h for hour and s for seconds.

Worked Case

Calculating Average Velocity

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

Strategy

We know that the displacement is 304 m north and the time is 180 s. We can utilize the formula for boilerplate velocity to solve the trouble.

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 exist 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, only the time interval 180 s has only two, and so the quotient should take only two significant figures.

Tips For Success

Annotation the way scalars and vectors are represented. In this book d represents distance and displacement. Similarly, v represents speed, and v represents velocity. A variable that is not bold indicates a scalar quantity, and a assuming variable indicates a vector quantity. Vectors are sometimes represented by minor arrows higher up the variable.

Teacher Support

Instructor Support

Use this problem to emphasize the importance of using the correct number of pregnant figures in calculations. Some students have a trend to include many digits in their final calculations. They incorrectly believe they are improving the accurateness of their answer by writing many of the digits shown on the calculator. Point out that doing this introduces errors into the calculations. In more complicated calculations, these errors can propagate and cause the final answer to be wrong. Instead, remind students to always carry ane or ii extra digits in intermediate calculations and to round the concluding answer to the right number of significant figures.

Worked Example

Solving for Displacement when Average Velocity and Fourth dimension are Known

Layla jogs with an average velocity of two.4 m/s e. What is her deportation after 46 seconds?

Strategy

We know that Layla's average velocity is 2.four m/s east, and the time interval is 46 seconds. We tin can rearrange the average velocity formula to solve for the deportation.

Discussion

The respond is near 110 yard e, which is a reasonable displacement for slightly less than a minute of jogging. A figurer shows the answer as 110.iv m. We chose to write the answer using scientific notation because we wanted to make it clear that we simply used two meaning figures.

Tips For Success

Dimensional analysis is a proficient way to determine whether you solved a problem correctly. Write the calculation using only units to be sure they match on reverse sides of the equal mark. In the worked case, you take
m = (m/s)(s). Since seconds is in the denominator for the average velocity and in the numerator for the time, the unit cancels out leaving only g and, of course, chiliad = yard.

Worked Example

Solving for Fourth dimension when Displacement and Average Velocity are Known

Phillip walks forth a direct path from his business firm to his school. How long volition it take him to get to school if he walks 428 1000 due west with an average velocity of ane.7 m/south due west?

Strategy

We know that Phillip's displacement is 428 m west, and his average velocity is 1.7 yard/due south due west. We tin can calculate the time required for the trip past rearranging the average velocity equation.

Discussion

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

Practice Problems

11 .

A trucker drives along a straight highway for 0.25 h with a displacement of 16 km south. What is the trucker's average velocity?

1. 4 km/h north

2. 4 km/h s

3. 64 km/h north

4. 64 km/h south

12 .

A bird flies with an average velocity of 7.v m/due south east from one branch to another in 2.4 south. Information technology then pauses before flying with an average velocity of 6.8 m/s east for 3.five s to some other branch. What is the bird'due south full displacement from its starting betoken?

1. 42 1000 westward
2. half dozen thousand west
3. half dozen m east
4. 42 m e

Virtual Physics

The Walking Man

In this simulation you will put your cursor on the man and move him first in one direction and then in the opposite direction. Keep the Introduction tab agile. Y'all can utilize the Charts tab later on yous larn almost graphing motility later in this chapter. Advisedly watch the sign of the numbers in the position and velocity boxes. Ignore the acceleration box for now. Run across if you can make the man's position positive while the velocity is negative. Then run across if y'all can practise the opposite.

Grasp Bank check

Which situation correctly describes when the moving man's position was negative but his velocity was positive?

1. Man moving toward 0 from left of 0
2. Human being moving toward 0 from right of 0
3. Man moving abroad from 0 from left of 0
4. Homo moving away from 0 from right of 0

Teacher Back up

Teacher Support

This is a powerful interactive animation, and it can be used for many lessons. At this indicate it tin can be used to evidence that displacement can be either positive or negative. Information technology can also show that when displacement is negative, velocity tin be either positive or negative. Later it can be used to evidence that velocity and acceleration tin have unlike signs. Information technology is strongly suggested that you keep students on the Introduction tab. The Charts tab can be used after students learn about graphing move afterward in this affiliate.

Check Your Agreement

13 .

2 runners traveling along the aforementioned straight path outset and stop their run at the same time. At the halfway marking, they have different instantaneous velocities. Is it possible for their boilerplate velocities for the entire trip to be the same?

1. Yes, considering average velocity depends on the net or total deportation.

2. Yes, considering boilerplate velocity depends on the total distance traveled.

3. No, considering the velocities of both runners must remain exactly the same throughout the journeying.

4. No, because the instantaneous velocities of the runners must remain the same at the midpoint but can vary at other points.

fourteen .

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

1. Average speed. Both are the aforementioned when the machine is traveling at a constant speed and changing direction.
2. Average speed. Both are the aforementioned when the speed is constant and the car does not alter its direction.
3. Magnitude of average velocity. Both are aforementioned when the machine is traveling at a constant speed.
4. Magnitude of average velocity. Both are same when the car does not change its direction.

15 .

Is it possible for average velocity to be negative?

1. Yes, if net displacement is negative.

2. Yes, if the object'southward management changes during motility.

3. No, considering average velocity describes only the magnitude and not the direction of motility.

4. No, because boilerplate velocity simply describes the magnitude in the positive direction of motion.

Teacher Back up

Teacher Support

Employ the Check Your Understanding questions to assess students' achievement of the sections learning objectives. If students are struggling with a specific objective, the Check Your Understanding will help identify which and direct students to the relevant content. Cess items in TUTOR will allow yous to reassess.

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Source: https://openstax.org/books/physics/pages/2-2-speed-and-velocity

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