Displacement, Velocity and Acceleration — Scalars and Vectors in 1D — Mathematics

Introduction

Kinematics is the study of how objects move, without asking why they move. Before building equations of motion or analysing real problems, the 9709 exam requires you to be precise about the language of motion. Two pairs of quantities are central:

Distance and Speed — scalar quantities (magnitude only).
Displacement, Velocity and Acceleration — vector quantities (magnitude and direction).

In one-dimensional motion the direction is simply "positive" or "negative" along a chosen line (e.g. to the right is positive, to the left is negative). Getting these definitions exactly right underpins every Mechanics question you will encounter.

Core Concept

Scalars vs Vectors

A scalar quantity is fully described by a single number (its magnitude). A vector quantity requires both a magnitude and a direction.

Scalar	Vector counterpart
Distance $d$	Displacement $s$
Speed $v$	Velocity $v$ (signed)
—	Acceleration $a$ (signed)

Because we work in one dimension only, direction is encoded simply by the sign of the number.

Distance and Displacement

Distance is the total length of path travelled, always $\geq 0$ .

Displacement is the change in position measured from a fixed reference point (the origin), in a specified positive direction. It can be negative, zero, or positive.

Example: A particle moves 5 m to the right, then 3 m to the left.

Distance travelled $= 5 + 3 = 8\ \text{m}$

Displacement from start $= 5 - 3 = +2\ \text{m}$ (to the right)

Speed and Velocity

Speed is the rate of change of distance with time — always $\geq 0$ .

Velocity is the rate of change of displacement with time — it carries a sign.

v = \frac{\Delta s}{\Delta t}

A negative velocity means the particle is moving in the negative direction (e.g. to the left if rightward is positive).

Acceleration

Acceleration is the rate of change of velocity with time:

a = \frac{\Delta v}{\Delta t}

Acceleration is also a vector in 1D: a positive value means velocity is increasing in the positive direction; a negative value means velocity is decreasing in the positive direction.

Deceleration

The term deceleration describes a situation where the speed is decreasing — i.e. the acceleration vector points opposite to the velocity vector. For example, if a particle moves in the positive direction ( $v > 0$ ) but has $a < 0$ , it is decelerating. Equally, if $v < 0$ and $a > 0$ , the particle is also decelerating (slowing down while moving left). Always interpret "deceleration" as a magnitude: a deceleration of $3\ \text{m s}^{-2}$ means $|a| = 3\ \text{m s}^{-2}$ acting against the motion.

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