Describing Motion

6.1 What Motion Means

An object is in motion when its position changes relative to a reference point over time. This definition seems simple, but it contains two important ideas: motion always involves a change in position, and that change is always measured relative to something else.

Motion is not an intrinsic property of an object. A passenger sitting in an airplane seat is motionless relative to the airplane, but moving at 900 km/h relative to the ground, and moving at a different speed still relative to the Sun. None of these descriptions is more “correct” than the others; they simply use different reference points. Physics accepts all of them as valid and provides the tools to translate between them.

The study of motion without asking what causes it is called kinematics. Kinematics describes the where, when, and how-fast of motion using position, time, velocity, and acceleration. The question of why objects move — what forces produce the motion — belongs to dynamics, which begins in Chapter 10. Chapters 6 through 9 are entirely kinematic: we describe motion without yet explaining it.

6.2 Position and Change in Position

Position is the location of an object measured relative to a chosen reference point called the origin, using a coordinate system. In one dimension, position is a single signed number with a unit: x = +3.5 m means the object is 3.5 meters in the positive direction from the origin. x = −2.0 m means the object is 2.0 meters in the negative direction.

Position is not the same as distance from the origin. Position is a signed quantity; it carries direction information. Distance from the origin is the magnitude of position and is always positive.

• Position: x = +3.5 m means 3.5 m in the positive direction.
• Position: x = −2.0 m means 2.0 m in the negative direction.

When an object moves, its position changes. The change in position is called displacement. Displacement is calculated by subtracting the initial position from the final position:

$\mathrm{\Delta x}=x_f-x_i$

6.3 Distance and Displacement

Distance and displacement sound similar but are fundamentally different quantities. Distance is the total length of the path traveled, regardless of direction. It is a scalar: always positive, always accumulating. Displacement is the straight-line change in position from start to finish. It is a vector: it has magnitude and direction, and it can be positive, negative, or zero.

The most important distinction: if an object returns to its starting point, its displacement is zero — regardless of how far it traveled. A runner completing one lap of a 400 m track has traveled a distance of 400 m but has a displacement of zero.

6.4 Reference Points and Reference Frames

A reference point, or origin, is the chosen location assigned coordinate zero. All positions are measured relative to this point. The choice of reference point is arbitrary; the physics does not change with the choice, but the numerical values of positions do.

A reference frame is a broader concept: it includes the reference point plus the coordinate axes and, crucially, the state of motion of the observer. Two observers in different states of motion — one standing still, one moving in a car — describe the same event with different numbers. Both descriptions are equally valid; they are simply expressed in different reference frames.

In everyday life, we almost always use Earth as our reference frame without thinking about it. In physics, we must be explicit about the reference frame whenever the motion of the observer matters.

6.5 One-Dimensional Motion

Most of Part II focuses on one-dimensional motion: motion along a single straight line. This is the simplest case and provides the foundation for understanding more complex motion in two and three dimensions.

In one-dimensional motion, we need only one coordinate, usually called x for horizontal motion or y for vertical motion, to specify position completely. The coordinate increases in the chosen positive direction and decreases in the negative direction. Because we are on a single axis, direction is fully encoded in the sign of the number.

Examples of one-dimensional motion include a car driving along a straight highway, a ball thrown straight up and falling back down, an elevator moving between floors, and a train moving along a straight track. In all of these, one coordinate is sufficient to describe the complete motion.

6.6 Motion Along a Straight Line

For motion along a straight line, we set up a number line aligned with the direction of motion. We choose an origin and a positive direction, then assign a numerical position to every point on the line.

The object’s position at any moment is a single number on this line. As time passes, that number changes. The sequence of positions over time completely describes the motion.

We often record positions in a data table, with columns for time and position. Each row is a snapshot of the object at one instant. Reading down the position column tells us whether the object moved in the positive direction, the negative direction, or not at all.

6.7 Positive and Negative Direction

The positive direction is the direction in which the coordinate axis increases. The negative direction is opposite. The choice of which direction is positive is made by the problem-solver, not by nature. However, once you make the choice, you must be consistent throughout the entire problem.

Common conventions: for horizontal motion, rightward is usually positive. For vertical motion, upward is usually positive, though some problems use downward as positive when an object falls. For a car on a road, the initial direction of motion is often chosen as positive.

The sign of a displacement tells you direction: positive means the object moved in the positive direction; negative means it moved in the negative direction. The sign of a position tells you which side of the origin the object is on: positive positions are on the positive side; negative positions are on the negative side.

6.8 Time Intervals and Elapsed Time

Time in physics is measured from a reference moment called t = 0, which is chosen by the problem-solver. It is usually the moment the motion begins or the moment an interesting event occurs, but any convenient reference point works.

A time interval, or elapsed time, is the duration between two moments: Δt = t_f − t_i. Like displacement, a time interval is a change: final value minus initial value. Time intervals are always positive or zero because time does not run backward in classical physics.

$\mathrm{\Delta t}=t_f-t_i$

Do not confuse the instant of time t with the time interval Δt. The instant t = 3 s tells you when something happened. The interval Δt = 3 s tells you how long a process lasted. These are different quantities and appear in different places in kinematic equations.

6.9 Motion Diagrams

A motion diagram is a visual representation of an object’s position at equally spaced time intervals. It shows a sequence of dots, each representing the object’s position at one instant, along the object’s path. Motion diagrams give an immediate visual sense of whether the object is moving fast or slow, speeding up or slowing down, and in what direction.

6.10 Position-Time Graphs

A position-time graph, or x-t graph, plots an object’s position on the vertical axis and time on the horizontal axis. It is one of the most information-rich representations of motion available, because its shape, slope, and features each carry physical meaning.

6.11 Uniform Motion vs. Changing Motion

Uniform motion means the object moves with constant velocity: same speed, same direction, the entire time. On an x-t graph, uniform motion produces a perfectly straight line. On a motion diagram, it produces equally spaced dots.

Changing motion means the object’s velocity is changing: it is speeding up, slowing down, or changing direction. On an x-t graph, changing motion produces a curve. On a motion diagram, it produces unequally spaced dots.

Uniform motion is the exception in the real world; most motion involves changing velocity. But understanding uniform motion first builds the conceptual foundation for understanding what acceleration means in Chapter 7 and what causes it in Chapter 10.

6.12 Describing Motion in Words, Tables, Graphs, and Diagrams

Physics uses four complementary representations to describe motion, and fluency in all four — and the ability to translate between them — is a core skill for this entire book.

Expert physicists move fluidly between these representations. A good problem-solving habit is to use at least two representations for any complex motion problem: draw a motion diagram to get the qualitative picture, then use a table or graph for quantitative analysis.

6.13 Common Mistakes When Describing Motion

Chapter Summary

Motion is a change in position relative to a reference point over time. Physics describes motion using four complementary tools: words, tables, graphs, and diagrams. The key quantities are position, displacement, distance, time, and time intervals.

• Position (x) is a signed number measured from a chosen origin. It requires a reference point, a positive direction, and a unit.
• Displacement (Δx = x_f − x_i) is the vector change in position: signed, direction-dependent, and path-independent.
• Distance is the total path length traveled: a scalar, always positive, and path-dependent.
• A reference frame includes a reference point and the state of motion of the observer. Motion descriptions are always relative to a reference frame.
• Time intervals (Δt = t_f − t_i) measure duration; they are distinct from the time coordinate t, which marks a specific instant.
• Motion diagrams show dot spacing: equal spacing means uniform motion; increasing gaps mean speeding up; decreasing gaps mean slowing down.
• On an x-t graph: straight line means constant velocity; curve means changing velocity; slope means velocity; steepness does not mean acceleration.
• Uniform motion has constant velocity, a straight x-t graph, and equally spaced dots. Changing motion has varying velocity, a curved x-t graph, and unequal dot spacing.

Key Terms

kinematics

The branch of physics that describes motion without reference to its causes; concerned with position, velocity, and acceleration.

motion

A change in position relative to a reference point over time.

position

The location of an object measured relative to a chosen origin using a coordinate system; a signed quantity with direction.

origin

The reference point assigned coordinate zero in a coordinate system.

displacement

The vector change in position from start to finish: Δx = x_f − x_i; signed, can be negative, and path-independent.

distance

The total length of the path traveled; a scalar, always positive, and path-dependent.

reference point

The chosen location assigned position zero; all positions are measured relative to it.

reference frame

A coordinate system plus the observer’s state of motion; defines the perspective from which motion is measured.

positive direction

The direction along an axis in which the coordinate increases; arbitrarily chosen but must be stated and held constant.

time interval

The duration between two moments: Δt = t_f − t_i; always positive in classical physics.

motion diagram

A representation of an object’s position at equally spaced time intervals, shown as a sequence of dots along the direction of motion.

position-time graph

A graph of position on the vertical axis versus time on the horizontal axis; the slope at any point equals the velocity.

uniform motion

Motion at constant velocity: equal displacements in equal time intervals; produces a straight line on an x-t graph.

changing motion

Motion in which velocity varies; produces a curved line on an x-t graph and unequal dot spacing in a motion diagram.

Concept Review Questions

Answer in complete sentences. No calculation required.

1. What does it mean to say that motion is relative? Give an example where an object is simultaneously at rest and moving, depending on the reference frame.
2. What is the difference between distance and displacement? Give an example where they have the same value, and one where they differ.
3. Why must you always state a reference point and positive direction before describing motion? What goes wrong if you do not?
4. What does a horizontal line on a position-time graph tell you about the object’s motion?
5. What does a negative slope on a position-time graph mean physically?
6. Explain the difference between a steep straight line and a curve on a position-time graph. What does each tell you about acceleration?
7. Describe what a motion diagram with decreasing dot spacing tells you about the object’s motion.
8. An athlete runs four laps of a 400 m track and returns to the start. What is the total distance? What is the displacement?
9. What is the difference between the time t and the time interval Δt? Give a physical example of each.
10. List the four representations of motion from Section 6.12. What is one advantage of each?

Motion Diagram Exercises

For each description, sketch a motion diagram showing the motion. Label the time interval between dots and indicate the positive direction with an arrow.

11. A car travels at constant speed to the right for 6 seconds.
12. A ball starts slowly and speeds up as it rolls down a ramp for 5 seconds.
13. A runner decelerates from top speed and comes to a stop over 4 seconds.
14. A toy car moves at constant speed for 3 seconds, stops for 2 seconds, then moves slowly in the opposite direction for 3 seconds.
15. A ball is thrown upward in the positive direction and rises for 2 seconds, then falls back down for 2 seconds.
16. For a motion diagram with equally spaced dots for 3 intervals, then gaps that increase for 3 more intervals, write a one-sentence description of the motion.

Position-Time Graph Practice

Use the following x-t graph description to answer questions 17–19. The graph shows four segments: A, a straight line with positive slope from t = 0 to t = 3 s; B, a horizontal line from t = 3 s to t = 5 s; C, a straight line with negative slope from t = 5 s to t = 9 s; and D, a curved line with slope increasing from t = 9 s to t = 12 s.

17. During which segments is the object moving in the positive direction? Negative direction? At rest?
18. During which segment is the object moving fastest? Slowest, excluding rest?
19. During which segment is the object accelerating, meaning velocity is changing?
20. Sketch an x-t graph for the following motion: an object starts at x = +10 m, moves to x = +4 m over 3 seconds at constant speed, then rests for 2 seconds, then moves to x = +14 m over 4 seconds at constant speed.
21. An x-t graph shows a straight line passing through the points (0 s, 2.0 m) and (5.0 s, 12.0 m). What is the slope? What are the units of the slope? What physical quantity does the slope represent?
22. Two objects A and B are shown on the same x-t graph. Object A starts at x = 0 and moves with a steep positive slope. Object B starts at x = +20 m and moves with a shallow negative slope. At what kind of point do their graphs intersect, and what does the intersection mean physically?

Distance and Displacement Problems

Show all work. Report displacement with sign and direction; report distance as a positive number.

23. A person walks 8.0 m east, then 5.0 m west. What is the total distance traveled? What is the displacement from start to finish?
24. A ball is thrown upward and rises 12.0 m before falling back to the thrower’s hand. What is the total distance traveled? What is the displacement?
25. A car starts at position x = +15.0 m and drives to position x = −8.0 m. What is the displacement? What is the distance traveled, assuming no reversal?
26. A hiker walks 3.0 km north, then 4.0 km east, then 3.0 km south. What is the total distance? What is the magnitude of the displacement? What is the direction of the displacement?
27. An ant crawls along the edge of a square tile of side 10 cm, completing three sides before stopping. How far did it travel? What is its displacement from start to finish?
28. A train starts at station A, defined as x = 0. It travels to station B at x = +42.0 km, then reverses to station C at x = +18.0 km. What is the distance traveled? What is the displacement from A to C? What is the displacement from B to C?

Applied Motion Problems

These problems combine position, displacement, reference frames, and graph interpretation.

29. Two students watch the same ball roll across a table. Student A sits at one end and defines the positive direction as away from her. Student B sits at the other end and defines the positive direction as away from him. The ball rolls from A toward B by 1.5 m. What is the displacement according to Student A? What is the displacement according to Student B? Do they disagree about the physics? Explain.
30. A dog runs 30 m due east in 6.0 s, then 20 m due west in 4.0 s. What is the total distance? What is the total displacement? Draw a position-time graph of the dog’s motion as a sketch, not precise scale.
31. A position-time graph shows a straight line from (0 s, 4.0 m) to (8.0 s, +12.0 m). What is the displacement? What is the slope? At what time does the object pass through x = 0?
32. An object’s position is recorded every 2 seconds: x = 0, 4, 8, 12, 8, 4, 0 m at t = 0, 2, 4, 6, 8, 10, 12 s. Describe the motion in words. What is the total distance traveled? What is the displacement from t = 0 to t = 12 s?
33. A passenger on a train moving at 25 m/s eastward throws a ball at 5 m/s toward the front of the train. What is the ball’s velocity relative to the train? Relative to the ground? If the passenger throws the ball toward the back instead, what is the ball’s velocity relative to the ground?

Chapter 6 Checkpoint

Before moving to Chapter 7, confirm you can do each of the following.