What Must You Do to Calculate a Meaningful Value of Distance From the Following Equation

Section Learning Objectives

By the terminate of this department, you will be able to do the following:

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

Teacher Support

Instructor Support

The learning objectives in this department will help your students chief the post-obit standards:

• (four) Science concepts. The educatee knows and applies the laws governing motility in a variety of situations. The educatee is expected to:
  • (B) describe and clarify motion in 1 dimension using equations with the concepts of distance, displacement, speed, average velocity, instantaneous velocity, and acceleration;
  • (F) identify and describe motion relative to unlike frames of reference.

Department Fundamental Terms

displacement	distance	kinematics	magnitude
position	reference frame	scalar	vector

Teacher Back up

Instructor Back up

[BL] [OL] First by asking what position is and how it is defined. Yous can use a toy automobile or other object. Then ask how they know the object has moved. Lead them to the idea of a defined starting point. Then bring in the concept of a numbered line as a way of quantifying motion.

[AL] Enquire students to depict the path of motion and emphasize that direction is a necessary component of a definition of movement. Ask the students to form pairs, and inquire each pair to come up with their own definition of motion. Then compare and discuss definitions equally a course. What components are necessary for a definition of motion?

Defining Motion

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

How exercise yous know something is moving? The location of an object at any particular fourth dimension is its position. More precisely, y'all need to specify its position relative to a convenient reference frame. Earth is oftentimes 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 Earth as a whole, while a professor's position could be described in terms of where she is in relation to the nearby white board. In other cases, we utilise reference frames that are non stationary but are in move relative to Earth. To describe the position of a person in an airplane, for case, we use the airplane, non Earth, as the reference frame. (Run into Figure two.2.) Thus, you can just know how fast and in what direction an object's position is changing against a groundwork of something else that is either not moving or moving with a known speed and direction. The reference frame is the coordinate system from which the positions of objects are described.

Figure 2.two Are clouds a useful reference frame for airplane passengers? Why or why not? (Paul Brennan, Public Domain)

Instructor Support

Teacher Support

[OL] [AL]Explain that the word kinematics comes from a Greek term meaning move. Information technology is related to other English words, such equally cinema (movies, or moving pictures) and kinesiology (the study of man motion).

Your classroom tin can be used as a reference frame. In the classroom, the walls are non moving. Your motion as you walk to the door, tin exist measured confronting the stationary background of the classroom walls. You tin besides tell if other things in the classroom are moving, such as your classmates entering the classroom or a book falling off a desk. You can also tell in what direction something is moving in the classroom. You might say, "The teacher is moving toward the door." Your reference frame allows you to decide not only that something is moving only as well the direction of motion.

You could as well serve as a reference frame for others' motility. If you remained seated as your classmates left the room, yous would measure their move away from your stationary location. If you and your classmates left the room together, then your perspective of their move would be change. You, as the reference frame, would exist moving in the aforementioned direction equally your other moving classmates. As you will learn in the Snap Lab, your description of move can be quite unlike when viewed from unlike reference frames.

Teacher Support

Teacher Support

[BL] [OL] You may desire to introduce the concept of a reference betoken every bit the starting point 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 not accelerating. Engage students in a discussion of how information technology is the difference in motion between the reference frame of the observer and the reference frame of the object that is of import in describing motion. The reference frames used in this affiliate might exist moving at a constant speed relative to each other, merely 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 motion is viewed. Demonstrate the difference by having one educatee stand up at the front of the course. Explicate that this educatee represents the background. Walk one time across the room between the pupil and the balance of the course. Ask the student and others in the class to describe the management of your motion. The class might describe your motion every bit to the right, but the student who is standing as a groundwork to your motion would describe the motility equally to the left. Conclude by reminding students that the reference frame is the viewpoint of the observer, not the groundwork.

[BL] Have students practise describing uncomplicated examples of motion in the class from different reference frames. For example, slide a book beyond a desk. Inquire students to depict its motion from their reference point, from the book's reference point, and from another student'southward reference signal.

Snap Lab

Looking at Motility from Two Reference Frames

In this action you will look at motion from 2 reference frames. Which reference frame is correct?

• Choose an open location with lots of infinite to spread out then there is less take chances of tripping or falling due to a collision and/or loose basketballs.

• 1 basketball

Procedure

1. Work with a partner. Stand a couple of meters away from your partner. Accept your partner plow to the side so that you are looking at your partner'southward profile. Have your partner begin bouncing the basketball while continuing in place. Describe the move of the brawl.
2. Side by side, take your partner again bounce the ball, but this time your partner should walk forward with the bouncing ball. You will remain stationary. Describe the brawl's movement.
3. Again have your partner walk forward with the billowy ball. This fourth dimension, you should move alongside your partner while continuing to view your partner's contour. Draw the ball's motion.
4. Switch places with your partner, and repeat Steps one–3.

Grasp Cheque

How do the different reference frames bear on how you describe the move of the ball?

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

Instructor Support

Instructor Back up

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

As students work through the lab, encourage lab partners to discuss their observations. In Steps ane and 3, students should find the ball movement straight up and straight downwardly. In Stride 2, students should observe the ball in a zigzag path abroad from the stationary observer.

Later the lab, lead students in discussing their observations. Ask them which reference frame is the correct ane. So emphasize that there is not a single right reference frame. All reference frames are every bit valid.

Links To Physics

History: Galileo's Ship

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

The idea that a description of motion depends on the reference frame of the observer has been known for hundreds of years. The 17^thursday-century astronomer Galileo Galilei (Figure 2.3) was one of the first scientists to explore this thought. Galileo suggested the following thought experiment: Imagine a windowless ship moving at a constant speed and direction forth a perfectly at-home ocean. Is at that place a way that a person within the ship tin can determine whether the ship is moving? You lot can extend this thought experiment by as well imagining a person standing on the shore. How tin a person on the shore decide whether the ship is moving?

Galileo came to an amazing conclusion. Only past looking at each other tin can a person in the send 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 depict the person on the country as moving by the ship. The person on shore would draw the ship and the person inside it as moving by. Galileo realized that observers moving at a abiding speed and direction relative to each other depict motion in the aforementioned style. Galileo had discovered that a description of motion is only meaningful if you specify a reference frame.

Imagine standing on a platform watching a train laissez passer past. According to Galileo's conclusions, how would your clarification of move and the clarification of move by a person riding on the train compare?

1. I would see the train every bit moving past me, and a person on the train would see me equally stationary.

2. I would come across the train equally moving by me, and a person on the train would come across me as moving by the train.

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

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

Distance vs. Displacement

Every bit we study the motion of objects, we must first be able to describe the object'due south position. Before your parent drives yous to school, the car is sitting in your driveway. Your driveway is the starting position for the auto. When yous reach your high school, the machine has changed position. Its new position is your schoolhouse.

Figure 2.iv Your full alter in position is measured from your house to your schoolhouse.

Physicists use variables to represent terms. We will utilise d to represent car'due south position. Nosotros will use a subscript to differentiate between the initial position, d ₀, and the final position, d _f. In addition, vectors, which nosotros will hash out later, will be in bold or will have an arrow above the variable. Scalars will be italicized.

Tips For Success

In some books, x or south 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 most two-dimensional motion, sometimes other subscripts will be used to describe horizontal position, d _x, or vertical position, d _y. Then, y'all might see references to d _0x and d _fy.

Now imagine driving from your house to a friend'due south house located several kilometers away. How far would yous bulldoze? The distance an object moves is the length of the path between its initial position and its final position. The distance you bulldoze to your friend's house depends on your path. Equally shown in Effigy two.5, distance is different from the length of a straight line betwixt ii points. The distance yous drive to your friend's house is probably longer than the straight line between the 2 houses.

Figure 2.v A brusque line separates the starting and catastrophe points of this motion, but the distance along the path of move is considerably longer.

We oft want to be more precise when we talk about position. The description of an object'due south move often includes more than just the distance it moves. For instance, if information technology is a v kilometer drive to schoolhouse, the altitude traveled is 5 kilometers. Afterwards dropping you off at school and driving dorsum home, your parent will have traveled a total altitude of 10 kilometers. The car and your parent will end upward in the same starting position in infinite. The internet change in position of an object is its displacement, or $\text{Δ}d\text{.}$ The Greek alphabetic character delta, $\text{Δ}$ , means modify in.

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

Teacher Support

Teacher Back up

Instructor Demonstration

Help students learn the divergence between distance and deportation by showing examples of movement.

1. As students sentry, walk direct across the room and have students gauge the length of your path.
2. Then, at aforementioned starting betoken, walk along a winding path to the aforementioned ending point.
3. Again, take students guess the length of your path.

Enquire—Which motility showed displacement? Which showed distance? Indicate out that the beginning motion shows displacement, and the second shows altitude along a path. In both cases, the starting and ending points were the same.

[OL] Be careful that students do not assume that initial position is always zero. Emphasize that although initial position is often nix, motion can start from whatsoever position relative to a starting point.

[Visual] Demonstrate positive and negative displacement by placing two meter sticks on the ground with their null marks end-to-end. Equally students watch, place a pocket-sized motorcar at the goose egg mark. Slowly movement the car to students' right a short distance and ask students what its displacement is. So move the car to the left of the zero mark. Point out that the machine now has a negative deportation.

Students volition larn more than about vectors and scalars later when they written report two-dimensional move. For at present, information technology is sufficient to introduce the terms and let students know that a vector includes information about direction.

[BL] Inquire students whether each of the following 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 big arrows cut from construction paper. Accept them use the arrows to place the magnitude (number or length of arrows) and direction of deportation. Emphasize that distance cannot be represented past arrows because distance does not include direction.

Snap Lab

Distance vs. Deportation

In this activity you volition compare distance and deportation. Which term is more than useful when making measurements?

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

Process

1. One student from each pair should stand with their back to the longest wall in the classroom. Students should stand up at least 0.v meters away from each other. Mark this starting point with a piece of masking record.
2. The 2nd pupil from each pair should stand facing their partner, most two to three meters abroad. Mark this point with a second piece of masking record.
3. Student pairs line up at the starting point along the wall.
4. The instructor turns on the music. Each pair walks back and forth from the wall to the second marked point until the music stops playing. Keep count of the number of times y'all walk across the floor.
5. When the music stops, mark your ending position with the third slice of masking tape.
6. Measure from your starting, initial position to your ending, final position.
7. Measure the length of your path from the starting position to the second marked position. Multiply this measurement past the full number of times you walked across the floor. Then add this number to your measurement from stride half-dozen.
8. Compare the two measurements from steps 6 and 7.

1. Which measurement is your total distance traveled?
2. Which measurement is your displacement?
3. When might yous want to use one 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 altitude to describe the total path betwixt starting and catastrophe points,and utilize deportation to describe the shortest path between starting and catastrophe points.

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

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

Teacher Support

Teacher Support

Choose a room that is large enough for all students to walk unobstructed. Brand sure the total path traveled is short enough that students can walk dorsum and forth across information technology multiple times during the course of a vocal. Take them measure out the distance between the two points and come up to a consensus. When students measure their deportation, make sure that they measure out forward from the direction they marked as the starting position. Later on they have completed the lab, have them discuss their results.

If you are describing only your bulldoze to schoolhouse, then the distance traveled and the displacement are the same—five kilometers. When you are describing the entire round trip, distance and deportation are different. When you depict distance, you simply include the magnitude, the size or amount, of the distance traveled. Yet, when you describe the deportation, you lot accept into account both the magnitude of the change in position and the direction of movement.

In our previous example, the car travels a full of ten kilometers, simply it drives five of those kilometers forrard toward school and five of those kilometers back in the opposite management. If we ascribe the forrad management a positive (+) and the opposite direction a negative (–), then the 2 quantities will abolish each other out when added together.

A quantity, such as altitude, that has magnitude (i.east., how large or how much) but does not take into business relationship direction is chosen a scalar. A quantity, such every bit displacement, that has both magnitude and direction is called a vector.

Lookout Physics

Vectors & Scalars

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

How does this video help you sympathize the difference between distance and deportation? Describe the differences between vectors and scalars using concrete quantities every bit examples.

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

2. Information technology explains that distance is a scalar and management is important, whereas displacement is a vector and it has no direction attached to information technology.

3. It explains that altitude is a scalar and information technology has no direction attached to it, whereas deportation is a vector and direction is important.

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

Instructor Support

Teacher Support

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

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

[AL] Discuss how the concept of direction might be important for the study of move.

Deportation Problems

Hopefully you now understand the conceptual divergence betwixt distance and displacement. Understanding concepts is half the boxing in physics. The other one-half is math. A stumbling block to new physics students is trying to wade through the math of physics while also 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'southward review and see if nosotros can make sense of deportation in terms of numbers and equations. You can calculate an object's displacement by 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 terminal position is the same as the initial position, then $\text{Δ}d=0$ .

To assign numbers and/or direction to these quantities, we need to define an axis with a positive and a negative direction. We as well need to define an origin, or O. In Figure 2.6, the axis is in a direct line with dwelling at zero and school in the positive direction. If we left abode and drove the reverse way from schoolhouse, motility would have been in the negative direction. We would have assigned information technology a negative value. In the round-trip drive, d _f and d ₀ were both at nix kilometers. In the one way trip to school, d _f was at 5 kilometers and d ₀ was at nada km. So, $\Delta d$ was 5 kilometers.

Tips For Success

You may place your origin wherever yous would like. You lot have to brand sure that you calculate all distances consistently from your zero and y'all ascertain one management equally positive and the other equally negative. Therefore, it makes sense to choose the easiest axis, direction, and zero. In the example higher up, we took dwelling to be zero because it allowed u.s. to avert having to interpret a solution with a negative sign.

Worked Example

Calculating Altitude and Displacement

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

Strategy

To solve this problem, we demand to observe the difference between the final position and the initial position while taking intendance to note the management on the axis. The final position is the sum of the two displacements, $\text{Δ}{d}_1$ and $\text{Δ}{d}_2$ .

Word

The displacement is negative because we chose east to be positive and west to be negative. We could also have described the displacement as i km west. When calculating displacement, the direction mattered, but when computing distance, the direction did not affair. The problem would work the same way if the trouble were in the due north–south or y-direction.

Tips For Success

Physicists like to use standard units so information technology is easier to compare notes. The standard units for calculations are chosen SI units (International Arrangement of Units). SI units are based on the metric system. The SI unit for displacement is the meter (grand), but sometimes y'all will see a problem with kilometers, miles, feet, or other units of length. If ane unit in a problem is an SI unit and some other is not, you will need to convert all of your quantities to the aforementioned system before you can acquit out the calculation.

Teacher Support

Teacher Support

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

Practice Problems

1 .

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

1. Deportation is -15 thousand and distance is -49 m.
2. Displacement is -15 m and distance is 49 m.
3. Displacement is xv grand and distance is -49 grand.
4. Displacement is 15 m and altitude is 49 thousand.

2 .

Tiana jogs 1.v km along a direct path and then turns and jogs 2.4 km in the opposite management. She and then turns back and jogs 0.7 km in the original direction. Permit Tiana's original management be the positive direction. What are the displacement and distance she jogged?

1. Displacement is 4.half dozen km,and altitude is -0.ii km.
2. Deportation is -0.2 km, and distance is 4.6 km.
3. Displacement is 4.6 km, and altitude is +0.two km.
4. Displacement is +0.2 km, and altitude is four.vi km.

Work In Physics

Mars Probe Explosion

Figure 2.vii The Mars Climate Orbiter disaster illustrates the importance of using the correct calculations in physics. (NASA)

Physicists make calculations all the fourth dimension, only they do non always get the right answers. In 1998, NASA, the National Aeronautics and Space Administration, launched the Mars Climate Orbiter, shown in Figure 2.7, a $125-one thousand thousand-dollar satellite designed to monitor the Martian atmosphere. It was supposed to orbit the planet and take readings from a prophylactic 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 ended upwardly flight into the Martian atmosphere. The probe disintegrated. It was one of the biggest embarrassments in NASA's history.

3 .

In 1999 the Mars Climate Orbiter crashed considering calculation were performed in English language units instead of SI units. At i signal the orbiter was just 187,000 feet above the surface, which was likewise shut to stay in orbit. What was the pinnacle of the orbiter at this time in kilometers? (Assume 1 meter equals 3.281 feet.)

1. 16 km

2. xviii km

3. 57 km

4. 614 km

Teacher Support

Instructor Support

The text characteristic describes a existent-life miscalculation made 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 only a certain amount of atmospheric pressure. The orbiter had to be close enough to the planet to take measurements and far plenty away that it could remain structurally audio. One way to teach this concept would be to choice an orbital distance from Mars and have the students calculate the distance of the path and the peak from the surface both in SI units and in English units. Ask why failure to convert might exist a problem.

Check Your Understanding

4 .

What does information technology mean when move is described equally relative?

1. It means that motion of any object is described relative to the movement of World.
2. Information technology means that motion of any object is described relative to the move of any other object.
3. Information technology means that motion is independent of the frame of reference.
4. It ways 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 y'all both view the motion from the same reference frame?

1. Yeah, we would both view the motion from the aforementioned reference point because both of us are at rest in Earth's frame of reference.

2. Aye, we would both view the motion from the same reference indicate because both of us are observing the motion from two points on the aforementioned straight line.

3. No, we would both view the movement from unlike reference points because move is viewed from two dissimilar points; the reference frames are similar but not the same.

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

6 .

What is the difference between distance and deportation?

1. Distance has both magnitude and direction, while deportation has magnitude but no direction.

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

3. Distance has magnitude simply no management, while displacement has only management.

4. At that place is no difference. Both altitude and deportation have magnitude and direction.

7 .

Which statement correctly describes a race car's distance traveled and magnitude of displacement during a one-lap motorcar race around an oval track?

1. The perimeter of the race track is the distance; the shortest altitude between the get-go line and the finish line is the magnitude of displacement.

2. The perimeter of the race track is the magnitude of displacement; the shortest distance betwixt the beginning and terminate line is the altitude.

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

4. The shortest distance between the beginning and the terminate line is the magnitude of the deportation vector.

viii .

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

1. Because Globe is continuously in movement; an object at rest on Earth volition be in motion when viewed from outer space.

2. Considering the position of a moving object can be divers only when there is a fixed reference frame.

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

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

Teacher Support

Instructor Support

Use the questions under Check Your Agreement to appraise students' accomplishment of the section's learning objectives. If students are struggling with a specific objective, the determinative assessment will help directly students to the relevant content.

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