2. Distance and Displacement

Learning outcomes
  • I can define distance as the total path traveled by an object.
  • I can define displacement as the change in position of an object.
  • I can distinguish between distance and displacement.
  • I can calculate distance traveled in simple situations.
  • I can determine the displacement of an object and describe its direction.

Introduction

When describing motion, scientists need more than just the words "moving" or "not moving." They need precise ways to describe how far an object has travelled and where it ends up. These ideas are represented by two related but different quantities: distance and displacement.

Although these terms are often used interchangeably in everyday conversation, they have very different meanings in physics. Understanding the difference between them is essential before studying speed, velocity, and acceleration. In this lesson, you will learn how to measure both distance and displacement and why direction is important when describing motion.

What is Distance?

Distance is the total length of the path travelled by an object.

Distance tells us how much ground an object has covered, regardless of the direction of travel.

Distance is a scalar quantity, meaning it has:

  • Magnitude only.
  • No direction.

Distance is always:

  • Positive.
  • Measured in units such as metres (m) or kilometres (km).

Examples:

  • A person walks 200 m to school.
  • A car travels 15 km to work.
  • A runner completes a 400 m race.

What is Displacement?

Displacement is the change in an object's position from its starting point to its ending point.

Displacement describes the shortest straight-line distance between the initial and final positions.

Unlike distance, displacement is a vector quantity, meaning it has:

  • Magnitude.
  • Direction.

Examples:

  • 50 m east
  • 12 km north
  • 5 m downward

Without a direction, displacement is incomplete.

Distance vs Displacement

The key difference is:

  • Distance measures the entire path travelled.
  • Displacement measures the straight-line change in position.

A journey may have:

  • A large distance.
  • A much smaller displacement.

Or even:

  • A non-zero distance.
  • A zero displacement.

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Comparing Distance and Displacement

Distance  Displacement
Total path travelled  Straight-line change in position
Scalar quantity  Vector quantity
Magnitude only  Magnitude and direction
Always positive  Can be positive, negative, or zero (depending on the chosen coordinate system)
Depends on the path taken   Depends only on the starting and ending positions

Example 1: Walking in a Straight Line

A student walks 30 m east.

Distance

30 m

Displacement

30 m east

Since the student travelled in a straight line without changing direction:

Distance = Displacement

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Example 2: Taking a Longer Route

A student walks:

  • 40 m east.
  • Then 30 m north.

Distance

 

Displacement

The displacement is the straight-line distance from the starting point to the ending point.

Using Pythagoras' Theorem:

\( \sqrt[]{40^2 + 30^2} = 50 \ m \)

Direction:

Approximately 37° north of east.

Here:

  • Distance = 70 m
  • Displacement = 50 m northeast

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Example 3: Returning to the Start

A runner jogs:

  • 100 m east.
  • Then 100 m west.

Distance

 

Displacement

The runner finishes exactly where they started.

Displacement:

0 m

Even though the runner travelled 200 m, there was no overall change in position.

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Why Direction Matters

Direction is essential when describing displacement.

Consider two students.

Student A:

  • Walks 20 m east.

Student B:

  • Walks 20 m west.

Both travel the same distance.

However, their displacements are different because the directions are opposite.

Direction allows scientists to describe motion more accurately.

Calculating Distance

When an object travels along several sections of a journey:

Distance = Total length of every section added together

Example:

A cyclist rides:

  • 3 km north.
  • 2 km east.
  • 4 km south.

Distance:

The total path travelled is 9 km.

Determining Displacement

To find displacement:

  1. Identify the starting position.
  2. Identify the ending position.
  3. Draw the shortest straight line between them.
  4. Determine its length.
  5. State its direction.

Unlike distance, displacement ignores the route taken.

Only the initial and final positions matter.

When are Distance and Displacement Equal?

Distance and displacement are equal only when:

  • The object moves in a straight line.
  • The object never changes direction.

Examples:

  • Walking directly across a field.
  • Driving straight along a road.
  • A lift moving directly upward.

If the path changes direction, the distance becomes greater than the displacement.

Everyday Examples

Distance and displacement appear in many everyday situations.

Examples include:

  • A runner completing laps around a track.
  • A delivery driver following city streets.
  • A hiker walking along winding trails.
  • An aircraft flying between airports.
  • A ship crossing the ocean.

Engineers, pilots, and GPS navigation systems often use displacement to determine the shortest route between two locations.

Why Understanding Distance and Displacement is Important

Distance and displacement are fundamental ideas in mechanics.

They are used to calculate:

  • Speed.
  • Velocity.
  • Acceleration.
  • Average speed.
  • Average velocity.

Understanding the difference prevents common mistakes when solving motion problems.

Real-World Connections

Many technologies rely on displacement rather than distance.

Examples include:

  • GPS navigation calculates your displacement to determine the quickest route.
  • Pilots plan direct flight paths between airports.
  • Engineers measure structural movement relative to an original position.
  • Scientists track the motion of satellites and planets using displacement vectors.

Recognising the difference between distance and displacement allows these systems to make accurate predictions.

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Worked Example

A student walks:

  • 60 m north
  • 40 m south

Question

Determine:

  1. The total distance travelled.
  2. The displacement.

Solution

Step 1: Calculate Distance

Distance = 100 m

Step 2: Calculate Displacement

The student finishes:

Displacement = 20 m north

Did You Know?

  • During a 400 m race, athletes travel a distance of 400 m, but because they finish where they started, their displacement is 0 m.
  • GPS systems often calculate the shortest displacement between two locations but must account for roads and obstacles when determining the actual distance to travel.
  • Astronauts aboard the International Space Station travel over 28,000 km every hour, but after each orbit they return close to their original position relative to Earth, illustrating the difference between distance travelled and displacement.

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Key Terms

Distance — The total length of the path travelled by an object.

Displacement — The straight-line change in position from an object's starting point to its ending point, including direction.

Scalar Quantity — A quantity that has magnitude only.

Vector Quantity — A quantity that has both magnitude and direction.

Position — The location of an object relative to a reference point.

Direction — The orientation of movement, such as north, south, east, or west.

Key Takeaways

  • Distance is the total path travelled, while displacement is the straight-line change in position.
  • Distance is a scalar quantity with magnitude only, whereas displacement is a vector quantity that includes both magnitude and direction.
  • Distance is always positive, while displacement depends on both the final position and the chosen direction.
  • Distance and displacement are equal only when an object travels in a straight line without changing direction.
  • Calculating distance involves adding the lengths of all parts of a journey, while determining displacement requires comparing the starting and ending positions.
  • Understanding the difference between distance and displacement provides the foundation for learning speed, velocity, and acceleration.