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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 1: KinematicsDisplacement, Velocity & AccelerationDisplacement

Displacement (College Board AP® Physics 1: Algebra-Based)

Study Guide

Leander Oates

Written by: Leander Oates

Reviewed by: Caroline Carroll

How to calculate displacement

Defining an object

  • The term object can be used to describe a wide range of items in physics, from subatomic particles to entire galaxies

  • When the term object is used to describe something, the assumption is that the following properties are ignored:

    • size

    • shape

    • internal configuration

  • The object may be treated as:

    • a point mass

    • a point charge

Position

  • Position is defined as:

An object's location in space, relative to a reference point

  • In math, the position of an object is measured relative to the origin of a coordinate system

Graph with x and y axes originating from (0,0). A red cross marks a position connected to the origin by a dotted line labeled "position vector."
In math, the position of a point is measured relative to the origin of a coordinate system

  • In physics, the position of an object, x, is measured relative to a reference point

  • The reference point could be a coordinate system or another object

    • For example, point A is located at the coordinate (5, 3)

    • Or, the car is located at 4 m to the left of the lamp post

  • The relative nature of an object's position gives it a directional component

  • This is why position is a vector quantity with both magnitude and direction

Displacement

  • Displacement is defined as:

The change in an object's position

  • Displacement describes the difference in position between an object's starting position and its finishing position, regardless of the route taken

  • Since displacement describes the movement of an object from its initial position to its final position, displacement is also a vector quantity with both magnitude and direction

  • Displacement is described algebraically using the following equation:

increment x space equals space x space minus space x subscript 0

  • Where:

    • increment x = displacement, measured in straight m

    • x = final position, measured in straight m

    • x subscript 0 = initial position, measured in straight m

  • Notice that the increment x notation literally means a change in position

Worked Example

On an x, y coordinate system, Object 1 moves from (1, 1) to (7, 1) to (3, 1). Object 2 moves from (4, 3) to (1, 3) to (7, 3) and Object 3 moves from (5, 5) to (9, 5) to (1, 5).

Which of the following statements is true?

A: open vertical bar increment x subscript 1 close vertical bar space greater than space open vertical bar increment x subscript 2 close vertical bar space greater than space open vertical bar increment x subscript 3 close vertical bar space

B: open vertical bar increment x subscript 1 close vertical bar space equals space open vertical bar increment x subscript 2 close vertical bar space equals space open vertical bar increment x subscript 3 close vertical bar

C: open vertical bar increment x subscript 2 close vertical bar space greater than space open vertical bar increment x subscript 1 close vertical bar space greater than space open vertical bar increment x subscript 3 close vertical bar

D: open vertical bar increment x subscript 3 close vertical bar space greater than open vertical bar increment x subscript 2 close vertical bar space greater than open vertical bar increment x subscript 1 close vertical bar

The correct answer is D

Answer:

Step 1: Analyze the scenario

  • The y coordinates for each object are constant, so the objects are moving in one dimension only

  • Object 1 moves in the positive direction from (1, 1) to (7, 1), then it moves in the negative direction from (7, 1) to (3, 1)

  • Object 2 moves in the negative direction from (4, 3) to (1, 3), then it moves in the positive direction from (1, 3) to (7, 3)

  • Object 3 moves in the positive direction from (5, 5) to (9, 5), then it moves in the negative direction from (9, 5) to (1, 5)

  • Recall that displacement is the change in an object's position from its initial position to its final position

Step 2: Sketch a diagram of the situation or calculate the displacement algebraically

A graph with x and y axes shows different horizontal vectors. Black arrows show the movement of the objects, blue arrows show the net movement from the starting positions to the final positions.

  • On the diagram:

    • The lower black arrows show the movement of each object from its initial position to its intermediate position

    • The upper black arrows show the movement of each object from its intermediate position to its final position

    • The blue arrows show the net movement of each object from its initial position to its final position; this is the object's displacement

  • Calculating algebraically using the equation

increment x space equals space x space minus space x subscript 0

  • Object 1

    • Initial position on x-axis, x subscript 0 space equals space 1

    • Final position on x-axis, x space equals space 3

increment x subscript 1 space equals space 3 space minus space 1 space equals space 2

  • Object 2

    • Initial position on x-axis, x subscript 0 space equals space 4

    • Final position on x-axis, x space equals space 7

increment x subscript 2 space equals space 7 space minus space 4 space equals space 3

  • Object 3

    • Initial position on x-axis, x subscript 0 space equals space 5

    • Final position on x-axis, x space equals space 1

increment x subscript 3 space equals space 1 space minus space 5 space equals space minus 4

Step 3: Review the answer options

  • The answer options describe the relative values for the magnitude of the displacement, open vertical bar increment x close vertical bar so the direction of the displacement can be ignored

  • The object with the greatest magnitude of displacement is Object 3

  • The object with the smallest magnitude of displacement is Object 1

4 space greater than space 3 space greater than space 2

  • Therefore:

open vertical bar increment x subscript 3 close vertical bar space greater than open vertical bar increment x subscript 2 close vertical bar space greater than open vertical bar increment x subscript 1 close vertical bar

  • This is answer option D

Examiner Tips and Tricks

Displacement is a straight line from the object's starting position, x subscript 0, to its final position, x, therefore, the vector arrow of displacement will always lie along a single plane. So, when displacement is expressed in terms of the object's starting position, the direction is already stated in the x subscript 0 term. Hence, vector notation is not required if the x subscript 0 term is present.

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    Author
    Reviewer
    Leander Oates

    Author: Leander Oates

    Expertise: Physics

    Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.

    Caroline Carroll

    Author: Caroline Carroll

    Expertise: Physics Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.