Distance and displacement

Overview

Welcome to a deep dive into one of the foundational concepts of mechanics: Distance and Displacement. While they might sound like the same thing in everyday language, in Physics, they are critically different. Understanding this difference is not just about memorising definitions; it's about unlocking the language of motion and forces. OCR examiners frequently test this distinction in Topic P2 (Forces), often in questions that combine graphical skills, mathematical calculations, and precise use of terminology. Candidates who can confidently distinguish between a scalar quantity (like distance) and a vector quantity (like displacement) are better equipped to tackle more advanced topics like velocity, acceleration, and momentum. This guide will equip you with the knowledge, techniques, and exam strategies to move beyond common pitfalls and answer any question on this topic with confidence and precision.

Key Concepts

Concept 1: Scalar vs. Vector — The Fundamental Difference

At the heart of this topic is the distinction between scalar and vector quantities. This is a concept that applies across all of Physics, so mastering it here is essential.

• A scalar quantity is defined by its magnitude (size) only. It tells you "how much" of something there is. Examples you'll encounter frequently are distance, speed, mass, energy, and temperature. If you say you walked 500 metres, you have stated a distance — a scalar value.

• A vector quantity is defined by both its magnitude AND its direction. It tells you not only "how much" but also "which way". Examples include displacement, velocity, force, and acceleration. To describe a displacement fully, you must say, for instance, "500 metres due East." Without the direction, the information is incomplete.

Examiner's Tip: A common way to lose marks is to treat displacement as a scalar. If a question asks for displacement, your final answer MUST have a direction. Credit is specifically awarded for this component.

Concept 2: Calculating Resultant Displacement

Often, an object undergoes more than one displacement (e.g., walking 30m North, then 40m East). The resultant displacement is the single displacement that would take the object from its starting point to its final point in a straight line. There are two key methods for finding this.

**1. Perpendicular Vectors (Pythagoras' Theorem - Higher & Foundation)**When two displacement vectors are at right angles (90°) to each other, they form a right-angled triangle. The resultant displacement is the hypotenuse of this triangle.

• Magnitude: You calculate the magnitude using Pythagoras' Theorem: a² + b² = c².
• Direction: You find the direction using trigonometry (SOH CAH TOA). You will need to calculate the angle relative to a starting direction (e.g., North or East) and, for full marks, convert this to a three-figure bearing.

**2. Non-Perpendicular Vectors (Scale Drawing - Higher & Foundation)**When vectors are not at right angles, you must use a scale drawing. Precision is key to gaining marks here.

• Step 1: State your scale. Write it clearly on your page (e.g., 1 cm = 5 m). This is a mark-able point.
• Step 2: Draw the first vector. Use a ruler and protractor to draw the first arrow to the correct length and in the correct direction.
• Step 3: Draw the second vector. From the tip of the first arrow (a 'tip-to-tail' diagram), draw the second vector to scale and in the correct direction.
• Step 4: Draw the resultant. The resultant is the arrow drawn from the start of the first vector to the tip of the second. Measure its length and angle.
• Step 5: Convert back. Use your scale to find the magnitude of the resultant displacement and state its direction as a bearing.

Concept 3: The Special Case of Circular Motion

A favourite exam question involves an object moving in a circle and returning to its starting point. Consider a runner on a 400m athletics track.

• Distance after one lap: The runner has covered 400m of ground. The distance is 400 m.
• Displacement after one lap: The runner is back exactly where they started. The straight-line distance and direction from start to finish is zero. The displacement is 0 m.

This highlights the difference perfectly. Do not confuse zero displacement with zero distance.

Mathematical & Scientific Relationships

• Pythagoras' Theorem (Must memorise): a² + b² = c²
  Used to find the magnitude of the resultant of two perpendicular vectors.

• Trigonometric Ratios (SOH CAH TOA) (Given on formula sheet):

  • sin(θ) = Opposite / Hypotenuse
  • cos(θ) = Adjacent / Hypotenuse
  • tan(θ) = Opposite / Adjacent
    Used to find the direction (angle) of a resultant vector.

• Circumference of a Circle (Given on formula sheet): C = 2πr
  Used to calculate the distance travelled in one full circle.

Podcast: Deep Dive into Distance & Displacement

For an audio breakdown of these concepts, including exam tips and a quick-fire quiz, listen to our dedicated 10-minute podcast episode.

Practical Applications

While there isn't a specific required practical for this topic, the principles are fundamental to many others. For example, in the Forces and Motion required practical, you will measure the time taken for a trolley to travel a certain distance to calculate its speed and acceleration. Understanding that the tape measure measures distance is a basic but crucial point. Similarly, navigation systems in cars or on your phone constantly calculate the most efficient route (a distance problem) but also show you the direct displacement to your destination ("as the crow flies"). Air traffic controllers rely entirely on vector displacement to manage aircraft safely, ensuring they maintain separation in three-dimensional space.

Unit Conversions

Examiners may try to catch you out with units. Be prepared to convert:

• Kilometres to metres: 1 km = 1000 m
• Centimetres to metres: 100 cm = 1 m
• Millimetres to metres: 1000 mm = 1 m
  Always convert all values to standard SI units (metres for distance/displacement) before you begin any calculation.