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Department Learning Objectives

By the stop of this section, yous will be able to do the following:

• Depict motion in different reference frames
• Ascertain distance and displacement, and distinguish between the two
• Solve problems involving distance and displacement

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Teacher Support

The learning objectives in this department will assist 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 ane dimension using equations with the concepts of distance, displacement, speed, boilerplate velocity, instantaneous velocity, and acceleration;
  • (F) identify and describe motion relative to different frames of reference.

Section Central Terms

displacement	distance	kinematics	magnitude
position	reference frame	scalar	vector

Teacher Support

Instructor Support

[BL] [OL] Start by request what position is and how it is defined. You can use a toy motorcar or other object. So ask how they know the object has moved. Lead them to the idea of a defined starting indicate. And then bring in the concept of a numbered line as a fashion of quantifying movement.

[AL] Ask students to depict the path of movement and emphasize that management is a necessary component of a definition of motion. Ask the students to grade pairs, and ask each pair to come with their own definition of motion. And so compare and talk over definitions as a class. What components are necessary for a definition of motion?

Defining Move

Our study of physics opens with kinematics—the study of motion without considering its causes. Objects are in motion everywhere you look. Everything from a lawn tennis game to a space-probe flyby of the planet Neptune involves movement. When you are resting, your heart moves claret through your veins. Fifty-fifty in inanimate objects, atoms are always moving.

How exercise you know something is moving? The location of an object at any item time is its position. More precisely, you need to specify its position relative to a convenient reference frame. Globe is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For case, a rocket launch would be described in terms of the position of the rocket with respect to World as a whole, while a professor's position could be described in terms of where she is in relation to the nearby white lath. In other cases, we utilise reference frames that are not stationary simply are in movement relative to Earth. To describe the position of a person in an airplane, for example, nosotros use the airplane, not Earth, as the reference frame. (See Figure 2.2.) Thus, you tin only know how fast and in what direction an object'due south position is irresolute against a background of something else that is either non moving or moving with a known speed and direction. The reference frame is the coordinate arrangement from which the positions of objects are described.

Figure 2.2 Are clouds a useful reference frame for airplane passengers? Why or why not? (Paul Brennan, Public Domain)

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Teacher Support

[OL] [AL]Explain that the word kinematics comes from a Greek term meaning motion. It is related to other English words, such as movie theater (movies, or moving pictures) and kinesiology (the study of human being motion).

Your classroom tin be used as a reference frame. In the classroom, the walls are not moving. Your move as you walk to the door, tin can be measured against the stationary background of the classroom walls. Y'all tin can also tell if other things in the classroom are moving, such as your classmates entering the classroom or a book falling off a desk-bound. You lot can as well tell in what management something is moving in the classroom. You might say, "The teacher is moving toward the door." Your reference frame allows you lot to determine non merely that something is moving only likewise the direction of move.

You could besides serve as a reference frame for others' movement. If you lot remained seated as your classmates left the room, you would measure their motility away from your stationary location. If you and your classmates left the room together, and then your perspective of their move would exist alter. You, every bit the reference frame, would be moving in the same direction as your other moving classmates. As you will learn in the Snap Lab, your description of motion can be quite different when viewed from different reference frames.

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Teacher Support

[BL] [OL] Yous may want to introduce the concept of a reference point equally the starting signal of motion. Relate this to the origin of a coordinate filigree.

[AL] Explain that the reference frames considered in this chapter are inertial reference frames, which means they are non accelerating. Engage students in a word of how it is the difference in motion between the reference frame of the observer and the reference frame of the object that is important in describing motility. The reference frames used in this chapter might be moving at a constant speed relative to each other, simply they are not accelerating relative to each other.

[BL] [OL][Visual] Misconception: Students may assume that a reference frame is a background of motion instead of the frame from which motility is viewed. Demonstrate the difference by having one student stand at the front of the form. Explicate that this student represents the background. Walk in one case beyond the room betwixt the student and the remainder of the course. Ask the student and others in the grade to draw the direction of your motion. The course might describe your motion every bit to the right, merely the pupil who is standing as a background to your motion would describe the motion as to the left. Conclude by reminding students that the reference frame is the viewpoint of the observer, not the background.

[BL] Have students practice describing simple examples of motility in the class from different reference frames. For example, slide a book across a desk-bound. Enquire students to describe its motion from their reference signal, from the volume's reference signal, and from another student'due south reference point.

Snap Lab

Looking at Motion from Ii Reference Frames

In this activity you will expect at motility from two reference frames. Which reference frame is correct?

• Cull an open location with lots of space to spread out so at that place is less adventure of tripping or falling due to a collision and/or loose basketballs.

• ane basketball game

Procedure

1. Work with a partner. Stand a couple of meters away from your partner. Take your partner plough to the side so that yous are looking at your partner's profile. Have your partner begin bouncing the basketball game while standing in identify. Draw the motion of the ball.
2. Next, have your partner again bounciness the ball, but this time your partner should walk forward with the billowy ball. You lot will remain stationary. Depict the ball'south motion.
3. Again have your partner walk forward with the bouncing ball. This time, you should move alongside your partner while continuing to view your partner's profile. Describe the brawl'southward motion.
4. Switch places with your partner, and repeat Steps 1–iii.

Grasp Cheque

How practise the different reference frames affect how you lot describe the move of the ball?

1. The motility of the ball is independent of the reference frame and is same for unlike reference frames.
2. The motion of the brawl is independent of the reference frame and is different for different reference frames.
3. The move of the brawl is dependent on the reference frame and is aforementioned for different reference frames.
4. The move of the ball is dependent on the reference frames and is different for different reference frames.

Teacher Support

Teacher Back up

Before students begin the lab, conform a location where pairs of students can have ample room to walk forward at least several meters.

As students work through the lab, encourage lab partners to discuss their observations. In Steps 1 and 3, students should observe the brawl move straight up and direct down. In Step 2, students should observe the ball in a zigzag path abroad from the stationary observer.

Afterward the lab, pb students in discussing their observations. Ask them which reference frame is the correct one. Then emphasize that there is non a single correct reference frame. All reference frames are every bit valid.

Links To Physics

History: Galileo's Ship

Figure 2.3 Galileo Galilei (1564–1642) studied motion and developed the concept of a reference frame. (Domenico Tintoretto)

The thought that a description of movement depends on the reference frame of the observer has been known for hundreds of years. The 17^thursday-century astronomer Galileo Galilei (Figure 2.iii) was 1 of the first scientists to explore this idea. Galileo suggested the post-obit thought experiment: Imagine a windowless ship moving at a constant speed and direction along a perfectly calm bounding main. Is in that location a way that a person inside the ship tin can make up one's mind whether the ship is moving? You can extend this thought experiment by also imagining a person standing on the shore. How tin a person on the shore determine whether the transport is moving?

Galileo came to an amazing conclusion. Only by looking at each other tin a person in the transport or a person on shore describe the motion of one relative to the other. In addition, their descriptions of motion would exist identical. A person inside the ship would describe the person on the country as moving past the transport. The person on shore would describe the ship and the person within it as moving past. Galileo realized that observers moving at a constant speed and management relative to each other draw movement in the same manner. Galileo had discovered that a description of move is but meaningful if you specify a reference frame.

Imagine continuing on a platform watching a railroad train laissez passer past. Co-ordinate to Galileo'south conclusions, how would your clarification of motility and the clarification of motility by a person riding on the train compare?

1. I would see the train as moving past me, and a person on the train would come across me as stationary.

2. I would encounter the train as moving past me, and a person on the train would encounter me as moving past the train.

3. I would see the railroad train every bit stationary, and a person on the train would meet me as moving past the railroad train.

4. I would see the train as stationary, and a person on the railroad train would besides see me every bit stationary.

Distance vs. Deportation

Equally we study the motion of objects, we must first be able to describe the object's position. Before your parent drives y'all to schoolhouse, the auto is sitting in your driveway. Your driveway is the starting position for the auto. When y'all accomplish your high school, the car has changed position. Its new position is your school.

Figure 2.4 Your total change in position is measured from your house to your school.

Physicists use variables to stand for terms. We will use d to represent car'due south position. We volition use a subscript to differentiate betwixt the initial position, d ₀, and the final position, d _f. In improver, vectors, which nosotros will hash out later, will exist in assuming or will have an arrow above the variable. Scalars will be italicized.

Tips For Success

In some books, x or s is used instead of d to draw position. In d ₀, said d naught, the subscript 0 stands for initial. When we begin to talk about 2-dimensional motion, sometimes other subscripts will be used to depict horizontal position, d ₁₀, or vertical position, d _y. So, you might see references to d _0x and d _fy.

Now imagine driving from your house to a friend'south house located several kilometers away. How far would y'all drive? The distance an object moves is the length of the path between its initial position and its terminal position. The distance y'all drive to your friend's house depends on your path. As shown in Figure 2.5, distance is dissimilar from the length of a straight line between ii points. The altitude you bulldoze to your friend's house is probably longer than the directly line between the two houses.

Figure 2.5 A short line separates the starting and ending points of this move, but the distance along the path of motion is considerably longer.

We often want to be more precise when nosotros talk about position. The description of an object's move oft includes more than than just the distance it moves. For example, if it is a five kilometer bulldoze to school, the distance traveled is 5 kilometers. After dropping y'all off at school and driving back abode, your parent will have traveled a total distance of 10 kilometers. The machine and your parent volition finish up in the same starting position in space. The net change in position of an object is its deportation, or $\text{Δ}d\text{.}$ The Greek alphabetic character delta, $\text{Δ}$ , means alter in.

Figure two.6 The total distance that your car travels is 10 km, merely the total displacement is 0.

Teacher Support

Teacher Back up

Instructor Demonstration

Help students larn the divergence between altitude and deportation by showing examples of motion.

1. As students watch, walk straight across the room and have students guess the length of your path.
2. And then, at aforementioned starting point, walk forth a winding path to the same ending signal.
3. Once again, have students estimate the length of your path.

Inquire—Which motility showed deportation? Which showed distance? Point out that the first move shows deportation, and the second shows distance along a path. In both cases, the starting and ending points were the same.

[OL] Be conscientious that students do not assume that initial position is always nothing. Emphasize that although initial position is often zero, motion tin start from any position relative to a starting point.

[Visual] Demonstrate positive and negative displacement by placing 2 meter sticks on the ground with their zilch marks end-to-terminate. Every bit students watch, place a small car at the nix mark. Slowly motion the car to students' right a short distance and enquire students what its displacement is. And so movement the car to the left of the zero mark. Bespeak out that the car now has a negative displacement.

Students will learn more virtually vectors and scalars later when they report two-dimensional movement. For now, it is sufficient to introduce the terms and allow students know that a vector includes information about direction.

[BL] Enquire students whether each of the post-obit is a vector quantity or a scalar quantity: temperature (scalar), force (vector), mass (scalar).

[OL] Inquire students to provide examples of vector quantities and scalar quantities.

[Kinesthetic] Provide students with large arrows cut from construction paper. Have them use the arrows to identify the magnitude (number or length of arrows) and management of displacement. Emphasize that distance cannot exist represented by arrows because distance does non include direction.

Snap Lab

Distance vs. Deportation

In this activity you lot will compare distance and displacement. Which term is more useful when making measurements?

• i recorded song bachelor on a portable device
• one record measure out
• three pieces of masking tape
• A room (like a gym) with a wall that is large and clear enough for all pairs of students to walk back and forth without running into each other.

Process

1. One educatee from each pair should stand with their back to the longest wall in the classroom. Students should stand at least 0.5 meters away from each other. Mark this starting point with a piece of masking tape.
2. The second educatee from each pair should stand up facing their partner, about 2 to three meters away. Marker this point with a second piece of masking record.
3. Student pairs line upwards at the starting point along the wall.
4. The instructor turns on the music. Each pair walks back and along from the wall to the 2d marked point until the music stops playing. Keep count of the number of times you walk across the floor.
5. When the music stops, mark your ending position with the third piece of masking record.
6. Mensurate from your starting, initial position to your ending, terminal position.
7. Measure the length of your path from the starting position to the 2d marked position. Multiply this measurement past the full number of times you walked across the floor. Then add together this number to your measurement from footstep 6.
8. Compare the two measurements from steps vi and 7.

1. Which measurement is your full distance traveled?
2. Which measurement is your deportation?
3. When might you want to use 1 over the other?

1. Measurement of the total length of your path from the starting position to the final position gives the altitude traveled, and the measurement from your initial position to your final position is the displacement. Use distance to describe the full path betwixt starting and catastrophe points,and use deportation to describe the shortest path between starting and catastrophe points.

2. Measurement of the full length of your path from the starting position to the terminal position is distance traveled, and the measurement from your initial position to your last position is deportation. Use distance to describe the shortest path between starting and ending points, and use deportation to depict the total path between starting and ending points.

3. Measurement from your initial position to your final position is distance traveled, and the measurement of the full length of your path from the starting position to the concluding position is deportation. Use altitude to depict the total path between starting and ending points, and use displacement to describe the shortest path between starting and ending points.

4. Measurement from your initial position to your final position is altitude traveled, and the measurement of the full length of your path from the starting position to the final position is displacement. Utilize altitude to describe the shortest path betwixt starting and ending points, and apply displacement to describe the total path betwixt starting and ending points.

Instructor Back up

Teacher Back up

Cull a room that is big enough for all students to walk unobstructed. Make sure the total path traveled is short plenty that students tin can walk back and forth across it multiple times during the course of a vocal. Have them measure the distance between the two points and come up to a consensus. When students measure their displacement, make sure that they measure forward from the direction they marked as the starting position. After they have completed the lab, take them discuss their results.

If you are describing just your drive to schoolhouse, then the distance traveled and the deportation are the same—5 kilometers. When yous are describing the unabridged round trip, distance and displacement are dissimilar. When you describe distance, y'all only include the magnitude, the size or amount, of the distance traveled. However, when you lot draw the deportation, you have into account both the magnitude of the alter in position and the direction of move.

In our previous example, the motorcar travels a total of x kilometers, only information technology drives five of those kilometers forward toward schoolhouse and five of those kilometers dorsum in the opposite direction. If we ascribe the forward direction a positive (+) and the reverse direction a negative (–), and then the 2 quantities volition cancel each other out when added together.

A quantity, such as distance, that has magnitude (i.eastward., how big or how much) but does not take into account direction is chosen a scalar. A quantity, such every bit displacement, that has both magnitude and direction is called a vector.

Picket Physics

Vectors & Scalars

This video introduces and differentiates between vectors and scalars. It also introduces quantities that we will be working with during the study of kinematics.

How does this video help y'all empathise the divergence between distance and displacement? Depict the differences betwixt vectors and scalars using concrete quantities equally examples.

1. It explains that distance is a vector and direction is important, whereas displacement is a scalar and it has no direction attached to it.

2. It explains that distance is a scalar and direction is important, whereas displacement is a vector and information technology has no direction fastened to information technology.

3. It explains that distance is a scalar and it has no management attached to it, whereas displacement is a vector and direction is important.

4. It explains that both distance and displacement are scalar and no directions are attached to them.

Teacher Support

Teacher Support

Define the concepts of vectors and scalars before watching the video.

[OL] [BL] Come up up with some examples of vectors and scalars and have the students classify each.

[AL] Discuss how the concept of direction might be of import for the report of motion.

Displacement Problems

Hopefully you now sympathise the conceptual deviation between distance and deportation. Understanding concepts is half the battle in physics. The other half is math. A stumbling block to new physics students is trying to wade through the math of physics while besides trying to empathize the associated concepts. This struggle may lead to misconceptions and answers that make no sense. Once the concept is mastered, the math is far less confusing.

So allow's review and see if we tin make sense of deportation in terms of numbers and equations. You can calculate an object's displacement past subtracting its original position, d₀ , from its final position d_f . In math terms that means

$$\text{Δ}d=d_{\text{f}}-d_0\text{.}$$

If the final position is the aforementioned every bit the initial position, and then $\text{Δ}d=0$ .

To assign numbers and/or direction to these quantities, we demand to define an centrality with a positive and a negative management. We likewise demand to ascertain an origin, or O. In Figure 2.half dozen, the axis is in a directly line with dwelling house at zero and schoolhouse in the positive direction. If we left domicile and collection the contrary mode from schoolhouse, move would take been in the negative management. We would take assigned it a negative value. In the circular-trip drive, d _f and d ₀ were both at cipher kilometers. In the i fashion trip to school, d _f was at 5 kilometers and d ₀ was at zero km. And then, $\Delta d$ was 5 kilometers.

Tips For Success

You lot may place your origin wherever you would like. You have to make sure that you lot calculate all distances consistently from your zero and you define one management as positive and the other as negative. Therefore, it makes sense to choose the easiest axis, direction, and zero. In the case above, we took home to be zero because information technology immune us to avoid having to translate a solution with a negative sign.

Worked Example

Calculating Distance and Displacement

A cyclist rides 3 km due west and and then turns around and rides 2 km e. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her deportation?

Strategy

To solve this problem, nosotros need to observe the difference betwixt the final position and the initial position while taking intendance to note the management on the axis. The terminal position is the sum of the two displacements, $\text{Δ}{d}_1$ and $\text{Δ}{d}_{\mathrm{ii}}$ .

Discussion

The displacement is negative because we chose eastward to be positive and west to be negative. We could also have described the displacement as ane km west. When calculating displacement, the direction mattered, but when calculating altitude, the direction did non matter. The problem would work the same style if the trouble were in the north–south or y-direction.

Tips For Success

Physicists like to use standard units so it is easier to compare notes. The standard units for calculations are called SI units (International System of Units). SI units are based on the metric system. The SI unit for displacement is the meter (m), but sometimes you will see a problem with kilometers, miles, feet, or other units of length. If one unit of measurement in a problem is an SI unit and another is not, you volition demand to convert all of your quantities to the same system earlier you tin carry out the adding.

Teacher Support

Instructor Support

Point out to students that the altitude for each segment is the absolute value of the displacement along a straight path.

Practice Bug

1 .

On an centrality in which moving from right to left is positive, what is the displacement and distance of a pupil who walks 32 m to the right and and so 17 m to the left?

1. Displacement is -15 g and altitude is -49 chiliad.
2. Displacement is -15 grand and altitude is 49 grand.
3. Displacement is 15 g and altitude is -49 m.
4. Displacement is 15 m and distance is 49 m.

2 .

Tiana jogs 1.five km along a straight path and and so turns and jogs 2.iv km in the reverse direction. She and then turns back and jogs 0.7 km in the original direction. Let Tiana'due south original direction be the positive management. What are the displacement and distance she jogged?

1. Displacement is iv.6 km,and distance is -0.2 km.
2. Displacement is -0.2 km, and distance is 4.half dozen km.
3. Displacement is four.6 km, and altitude is +0.2 km.
4. Deportation is +0.ii km, and altitude is iv.half dozen km.

Work In Physics

Mars Probe Explosion

Figure 2.7 The Mars Climate Orbiter disaster illustrates the importance of using the right calculations in physics. (NASA)

Physicists make calculations all the time, just they do not always go the correct answers. In 1998, NASA, the National Aeronautics and Space Administration, launched the Mars Climate Orbiter, shown in Figure 2.7, a $125-million-dollar satellite designed to monitor the Martian atmosphere. It was supposed to orbit the planet and take readings from a safety distance. The American scientists made calculations in English units (feet, inches, pounds, etc.) and forgot to convert their answers to the standard metric SI units. This was a very costly mistake. Instead of orbiting the planet as planned, the Mars Climate Orbiter concluded up flying into the Martian temper. The probe disintegrated. Information technology was ane of the biggest embarrassments in NASA's history.

3 .

In 1999 the Mars Climate Orbiter crashed because calculation were performed in English units instead of SI units. At 1 point the orbiter was only 187,000 feet above the surface, which was too close to stay in orbit. What was the height of the orbiter at this time in kilometers? (Presume one meter equals 3.281 feet.)

1. xvi km

2. xviii km

3. 57 km

4. 614 km

Teacher Back up

Teacher Support

The text feature describes a existent-life miscalculation fabricated by astronomers at NASA. In this case, the Mars Climate Orbiter's orbit needed to be calculated precisely because its machinery was designed to withstand merely a certain amount of atmospheric pressure level. The orbiter had to be close enough to the planet to accept measurements and far enough away that it could remain structurally audio. One way to teach this concept would be to choice an orbital altitude from Mars and have the students calculate the altitude of the path and the elevation from the surface both in SI units and in English units. Enquire why failure to convert might be a problem.

Bank check Your Understanding

4 .

What does information technology mean when movement is described as relative?

1. Information technology ways that move of any object is described relative to the movement of Earth.
2. It means that motion of whatever object is described relative to the movement of whatever other object.
3. It means that movement is independent of the frame of reference.
4. It means that motion depends on the frame of reference selected.

5 .

If you and a friend are standing side-by-side watching a soccer game, would you both view the motion from the same reference frame?

1. Yes, nosotros would both view the motion from the same reference point considering both of us are at residual in Earth's frame of reference.

2. Yes, we would both view the motility from the aforementioned reference point because both of united states of america are observing the motility from two points on the same straight line.

3. No, nosotros would both view the motion from different reference points because motion is viewed from two unlike points; the reference frames are like but not the same.

4. No, we would both view the move from dissimilar reference points because response times may be unlike; so, the motility observed by both of united states would be different.

half dozen .

What is the difference between distance and displacement?

1. Altitude has both magnitude and management, while displacement has magnitude but no direction.

2. Distance has magnitude simply no direction, while displacement has both magnitude and direction.

3. Distance has magnitude but no direction, while displacement has just direction.

4. There is no deviation. Both altitude and displacement have magnitude and direction.

7 .

Which statement correctly describes a race car's distance traveled and magnitude of deportation during a one-lap machine race around an oval rails?

1. The perimeter of the race rail is the distance; the shortest distance between the start line and the finish line is the magnitude of displacement.

2. The perimeter of the race runway is the magnitude of displacement; the shortest distance between the start and end line is the distance.

3. The perimeter of the race track is both the distance and magnitude of displacement.

4. The shortest distance between the get-go and the finish line is the magnitude of the displacement vector.

viii .

Why is it important to specify a reference frame when describing motion?

1. Because Earth is continuously in motion; an object at rest on World will be in motion when viewed from outer infinite.

2. Because the position of a moving object tin be defined simply when at that place is a stock-still reference frame.

3. Because motion is a relative term; it appears differently when viewed from different reference frames.

4. Because movement is always described in Earth's frame of reference; if another frame is used, it has to be specified with each situation.

Teacher Support

Teacher Support

Use the questions under Check Your Understanding to appraise students' achievement of the department'due south learning objectives. If students are struggling with a specific objective, the formative assessment will assist direct students to the relevant content.

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Source: https://openstax.org/books/physics/pages/2-1-relative-motion-distance-and-displacement

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