Relative Motion, Distance, and Displacement, Motion in One Dimension | Physics

Learning objectives

Section Key Terms

displacement	distance	kinematics	magnitude
position	reference frame	scalar	vector

Defining Motion

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 tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects, atoms are always moving.

How do you know something is moving? The location of an object at any particular time is its position. More precisely, you need to specify its position relative to a convenient reference frame. Earth 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 example, 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 use reference frames that are not stationary but are in motion relative to Earth. To describe the position of a person in an airplane, for example, we use the airplane, not Earth, as the reference frame. (See Figure_02_01_Airplane.) Thus, you can only know how fast and in what direction an object's position is changing against a background 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.

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

Original figure description: A photograph of an airplane flying in an upward direction is shown. The sky is partly cloudy.

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

Your classroom can be used as a reference frame. In the classroom, the walls are not moving. Your motion as you walk to the door, can be measured against the stationary background of the classroom walls. You 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. 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 determine not only that something is moving but also the direction of motion.

You could also serve as a reference frame for others’ movement. If you remained seated as your classmates left the room, you would measure their movement away from your stationary location. If you and your classmates left the room together, then your perspective of their motion would be change. You, as 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.

Teacher Support

[BL][OL] You may want to introduce the concept of a reference point as the starting point of motion. Relate this to the origin of a coordinate grid.

[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 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 motion. The reference frames used in this chapter might be moving at a constant speed relative to each other, but 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 student stand at the front of the class. Explain that this student represents the background. Walk once across the room between the student and the rest of the class. Ask the student and others in the class to describe the direction of your motion. The class might describe your motion as to the right, but the student 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 motion in the class from different reference frames. For example, slide a book across a desk. Ask students to describe its motion from their reference point, from the book's reference point, and from another student's reference point.

Looking at Motion from Two Reference Frames

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

Choose an open location with lots of space to spread out so there is less chance of tripping or falling due to a collision and/or loose basketballs.

1 basketball

Procedure

Work with a partner. Stand a couple of meters away from your partner. Have your partner turn to the side so that you are looking at your partner’s profile. Have your partner begin bouncing the basketball while standing in place. Describe the motion of the ball.
Next, have your partner again bounce the ball, but this time your partner should walk forward with the bouncing ball. You will remain stationary. Describe the ball's motion.
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 ball's motion.
Switch places with your partner, and repeat Steps 1–3.

How do the different reference frames affect how you describe the motion of the ball?

The motion of the ball is independent of the reference frame and is same for different reference frames.
The motion of the ball is independent of the reference frame and is different for different reference frames.
The motion of the ball is dependent on the reference frame and is same for different reference frames.
The motion of the ball is dependent on the reference frames and is different for different reference frames.

History: Galileo's Ship

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Original figure description: A painting of Galileo Galilei is shown.

Galileo Galilei (1564–1642) studied motion 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^th-century astronomer Galileo Galilei (Figure_02_01_Galileo) was one of the first scientists to explore this idea. Galileo suggested the following thought experiment: Imagine a windowless ship moving at a constant speed and direction along a perfectly calm sea. Is there a way that a person inside the ship can determine whether the ship is moving? You can extend this thought experiment by also imagining a person standing on the shore. How can a person on the shore determine whether the ship is moving?

Galileo came to an amazing conclusion. Only by looking at each other can a person in the ship or a person on shore describe the motion of one relative to the other. In addition, their descriptions of motion would be symmetric or opposite. A person inside the ship would describe the person on the land as moving past the ship. The person on shore would describe the ship and the person inside it as moving past. Galileo realized that observers moving at a constant speed and direction relative to each other describe motion in the same way. Galileo had discovered that a description of motion is only meaningful if you specify a reference frame.

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Distance vs. Displacement

As we study the motion of objects, we must first be able to describe the object’s position. Before your parent drives you to school, the car is sitting in your driveway. Your driveway is the starting position for the car. When you reach your high school, the car has changed position. Its new position is your school.

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

Original figure description: A drawing is shown of a house on the left facing a school on the right. A vector between points from the house to the school is labeled d o on the left and d f on the right. A car is shown driving from the house to the school.

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

Physicists use variables to represent terms. We will use d to represent car’s position. We will use a subscript to differentiate between the initial position, d₀, and the final position, d_f. In addition, vectors, which we will discuss later, will be in bold or will have an arrow above the variable. Scalars will be italicized.

Now imagine driving from your house to a friend's house located several kilometers away. How far would you drive? The distance an object moves is the length of the path between its initial position and its final position. The distance you drive to your friend's house depends on your path. As shown in Figure_02_01_comic, distance is different from the length of a straight line between two points. The distance you drive to your friend's house is probably longer than the straight line between the two houses.

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

Original figure description: A diagram is shown of a house on the left labeled home and another house on the right labeled Friend's house. The driveways of the houses are facing each other and a curvy road connects the driveways. A car is on the curvy road about half-way between the two houses.

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

We often want to be more precise when we talk about position. The description of an object’s motion often includes more than just the distance it moves. For instance, if it is a five kilometer drive to school, the distance traveled is 5 kilometers. After dropping you off at school and driving back home, your parent will have traveled a total distance of 10 kilometers. The car and your parent will end up in the same starting position in space. The net change in position of an object is its displacement, or $Δ d .$ The Greek letter delta, $Δ$ , means change in.

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

Original figure description: A drawing is shown of a house on the left facing a school on the right. The distance between the two is labeled five miles. A double-arrow vector between the house and the school is labeled do on the left and df on the right. The equation change in d total equals zero is shown below the vector.

The total distance that your car travels is 10 km, but the total displacement is 0.

Teacher Support

Help students learn the difference between distance and displacement by showing examples of motion.

As students watch, walk straight across the room and have students estimate the length of your path.
Then, at same starting point, walk along a winding path to the same ending point.
Again, have students estimate the length of your path.

Ask—Which motion showed displacement? Which showed distance? Point out that the first motion shows displacement, and the second shows distance 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 zero, motion can start from any position relative to a starting point.

[Visual] Demonstrate positive and negative displacement by placing two meter sticks on the ground with their zero marks end-to-end. As students watch, place a small car at the zero mark. Slowly move the car to students' right a short distance and ask students what its displacement is. Then move the car to the left of the zero mark. Point out that the car now has a negative displacement.

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

[BL] Ask students whether each of the following is a vector quantity or a scalar quantity: temperature (scalar), force (vector), mass (scalar).

[OL] Ask 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 direction of displacement. Emphasize that distance cannot be represented by arrows because distance does not include direction.

Distance vs. Displacement

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

1 recorded song available on a portable device
1 tape measure
3 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.

Procedure

One student 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.
The second student from each pair should stand facing their partner, about two to three meters away. Mark this point with a second piece of masking tape.
Student pairs line up at the starting point along the wall.
The teacher 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 you walk across the floor.
When the music stops, mark your ending position with the third piece of masking tape.
Measure from your starting, initial position to your ending, final position.
Measure the length of your path from the starting position to the second marked position. Multiply this measurement by the total number of times you walked across the floor. Then add this number to your measurement from step 6.
Compare the two measurements from steps 6 and 7.

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If you are describing only your drive to school and the route is a straight line, then the distance traveled and the displacement are the same—5 kilometers. When you are describing the entire round trip, distance and displacement are different. When you describe distance, you only include the magnitude, the size or amount, of the distance traveled. However, when you describe the displacement, you take into account both the magnitude of the change in position and the direction of movement.

In our previous example, the car travels a total of 10 kilometers, but it drives five of those kilometers forward toward school and five of those kilometers back in the opposite direction. If we ascribe the forward direction a positive (+) and the opposite direction a negative (–), then the two quantities will cancel each other out when added together.

A quantity, such as distance, that has magnitude (i.e., how big or how much) and sometimes a sign (e.g., electric charge, temperature in Celsius, or component of a vector) but does not take into account direction is called a scalar. A quantity, such as displacement, that has both magnitude and direction is called a vector. A vector with magnitude zero is a special case of a vector that has no direction.

Displacement Problems

Hopefully you now understand the conceptual difference between distance and displacement. 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 also trying to understand 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 let’s review and see if we can make sense of displacement 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

Δ d = d_{f} - d_{0} .

If the final position is the same as the initial position, then $Δ d = 0$ .

To assign numbers and/or direction to these quantities, we need to define an axis with a positive and a negative direction. We also need to define an origin, or O. In Figure_02_01_arrow, the axis is in a straight line with home at zero and school in the positive direction. If we left home and drove the opposite way from school, motion would have been in the negative direction. We would have assigned it a negative value. In the round-trip drive, d_f and d₀ were both at zero kilometers. In the one way trip to school, d_f was at 5 kilometers and d₀ was at zero km. So, $Δ d$ was 5 kilometers.

Calculating Distance and Displacement

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

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

Original figure description: A segment is shown, divided into two sections. The first section is labeled change in d two equals plus twelve kilometers and has a right pointing arrow, ending at a label df. The second section is labeled do at the end of it. A left-pointing arrow is above the entire segment and is labeled change in d one equals negative three kilometers. The right of the segment, a left-pointing arrow is shown with a W and a negative sign on the left side and E and a positive sign on the right side. The label do is in the center.

Solution

Displacement: The rider’s displacement is $Δ d = d_{f} - d_{0} = - 1 km$ .
Distance: The distance traveled is 3 km + 2 km = 5 km.
The magnitude of the displacement is 1 km.

Discussion

The displacement is negative because we chose east to be positive and west to be negative. We could also have described the displacement as 1 km west. When calculating displacement, the direction mattered, but when calculating distance, the direction did not matter. The problem would work the same way if the problem were in the north–south or y-direction.

Practice Problems

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

Displacement is -15 m and distance is -49 m.
Displacement is -15 m and distance is 49 m.
Displacement is 15 m and distance is -49 m.
Displacement is 15 m and distance is 49 m.

Tiana jogs 1.5 km along a straight path and then turns and jogs 2.4 km in the opposite direction. She then turns back and jogs 0.7 km in the original direction. Let Tianas original direction be the positive direction. What are the displacement and distance she jogged?

Displacement is 4.6 km,and distance is -0.2 km.
Displacement is -0.2 km, and distance is 4.6 km.
Displacement is 4.6 km, and distance is +0.2 km.
Displacement is +0.2 km, and distance is 4.6 km.

Mars Probe Explosion

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Original figure description: An image of the Mars Climate Orbiter is shown hovering above Mars.

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

Physicists make calculations all the time, but they do not always get the right answers. In 1998, NASA, the National Aeronautics and Space Administration, launched the Mars Climate Orbiter, shown in Figure_02_01_Mars, a $125-million-dollar satellite designed to monitor the Martian atmosphere. It was supposed to orbit the planet and take readings from a safe 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 up flying into the Martian atmosphere. The probe disintegrated. It was one of the biggest embarrassments in NASA’s history.

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Check Your Understanding

What does it mean when motion is described as relative?

It means that motion of any object is described relative to the motion of Earth.
It means that motion of any object is described relative to the motion of any other object.
It means that motion is independent of the frame of reference.
It means that motion depends on the frame of reference selected.

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Section Summary

A description of motion depends on the reference frame from which it is described.
The distance an object moves is the length of the path along which it moves.
Displacement is the difference in the initial and final positions of an object.

Key Equations

Displacement

Δ d = d_{f} - d_{0}

Concept Items

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In which example would you be correct in describing an object in motion while your friend would also be correct in describing that same object as being at rest?

You are driving a car toward the east and your friend drives past you in the opposite direction with the same speed. In your frame of reference, you will be in motion. In your friends frame of reference, you will be at rest.
You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, you will be in motion. In your friends frame of reference, you will be at rest.
You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, your friend will be moving toward the west. In your friends frame of reference, he will be at rest.
You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, your friend will be moving toward the east. In your friends frame of reference, he will be at rest.

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Critical Thinking

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Problems

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Billy drops a ball from a height of 1 m. The ball bounces back to a height of 0.8 m, then bounces again to a height of 0.5 m, and bounces once more to a height of 0.2 m. Up is the positive direction. What are the total displacement of the ball and the total distance traveled by the ball?

The displacement is equal to –4 m and the distance is equal to 4 m.
The displacement is equal to –1 m and the distance is equal to 1 m
The displacement is equal to 4 m and the distance is equal to 1 m.
The displacement is equal to –1 m and the distance is equal to 4 m.

Test Prep Multiple Choice

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If a space shuttle orbits Earth once, what is the shuttles distance traveled and displacement?

Distance and displacement both are zero.
Distance is circumference of the circular orbit while displacement is zero.
Distance is zero while the displacement is circumference of the circular orbit.
Distance and displacement both are equal to circumference of the circular orbit.

Test Prep Short Answer

While standing on a sidewalk facing the road, you see a bicyclist passing by toward your right. In the reference frame of the bicyclist, in which direction are you moving?

in the same direction of motion as the bicyclist
in the direction opposite the motion of the bicyclist
stationary with respect to the bicyclist
in the direction of velocity of the bicyclist

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Test Prep Extended Response

Find the distance traveled from the starting point for each path.

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

Original figure description: A line begins at 0 and extends to the right. 2, 4, 6, 8, 10, and 12 are marked on the line, and it is titled displacement x (m). a green line A extends from 0 to the right. A red line c extends from two to the right, and makes an ess pattern at 10 where it goes back to 8, then reverses direction again and extends to infinity. An orange line D extends from 9 to the left, then reverses direction at 3. A purple line B extends from 12 to the left.

Which path has the maximum distance?

The distance for Path A is 6 m, Path B is 4 m, Path C is 12 m and for Path D is 7 m. The net displacement for Path A is 7 m, Path B is –4m, Path C is 8 m and for Path D is –5m. Path C has maximum distance and it is equal to 12 meters.
The distance for Path A is 6 m, Path B is 4 m, Path C is 8 m and for Path D is 7 m. The net displacement for Path A is 6 m, Path B is –4m, Path C is 12 m and for Path D is –5 m. Path A has maximum distance and it is equal to 6 meters.
The distance for Path A is 6 m, Path B is 4 m, Path C is 12 m and for Path D is 7 m. The net displacement for Path A is 6 m, Path B is –4 m, Path C is 8 m and for Path D is –5 m. Path C has maximum distance and it is equal to 12 meters.
The distance for Path A is 6 m, Path B is –4 m, Path C is 12 m and for Path D is –5 m. The net displacement for Path A is 7 m, Path B is 4 m, Path C is 8 m and for Path D is 7 m. Path A has maximum distance and it is equal to 6 m.

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Glossary

displacement: the change in position of an object against a fixed axis
distance: the length of the path actually traveled between an initial and a final position
kinematics: the study of motion without considering its causes
magnitude: size or amount
position: the location of an object at any particular time
reference frame: a coordinate system from which the positions of objects are described
scalar: a quantity that has magnitude (and possibly sign) but no direction
vector: a quantity that has both magnitude and direction

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