Define displacement, velocity and acceleration for motion in a straight line and interpret their signs

What this dot point is asking

SEAB wants you to understand the three quantities that describe motion in a straight line: displacement, velocity and acceleration, what each means, and how their signs tell you about direction. You also need to see how they are connected as rates of change, which sets up the calculus methods in the next two dot points. The key idea is that these quantities are directed: their signs carry meaning.

The answer

Displacement

Displacement $s$ is the position of the particle measured from a fixed origin, including direction. It is usually a function of time, $s(t)$. A positive displacement means the particle is on the positive side of the origin; a negative displacement means the negative side. Displacement is not the same as distance travelled: a particle that goes out and comes back has zero displacement but a non-zero distance.

Velocity

Velocity $v$ is the rate of change of displacement with respect to time, so it measures how fast and in which direction the position is changing. The sign of velocity gives the direction of motion:

• $v > 0$: moving in the positive direction,
• $v < 0$: moving in the negative direction,
• $v = 0$: momentarily at rest.

Speed is the magnitude of velocity (its size without the sign).

Acceleration

Acceleration $a$ is the rate of change of velocity with respect to time. It tells you how the velocity is changing:

• $a > 0$: velocity is increasing,
• $a < 0$: velocity is decreasing,
• $a = 0$: velocity is constant.

A particle speeds up when velocity and acceleration have the same sign, and slows down when they have opposite signs.

How they connect

The chain is: displacement, then velocity (its rate of change), then acceleration (the rate of change of velocity). Going one way (displacement to velocity to acceleration) is differentiation; going back the other way is integration. This is exactly why the calculus strand and kinematics are taught together.

Examples in context

Example 1. A ball thrown upward. On the way up the velocity is positive but the acceleration (gravity) is negative, so the ball slows; at the top the velocity is momentarily zero while the acceleration continues, then the ball speeds up downward with velocity and acceleration both negative. The signs tell the whole story.

Example 2. A car braking. A car moving forward (positive velocity) that brakes has a negative acceleration, opposite in sign to the velocity, so it slows down. Reading the signs of velocity and acceleration tells you immediately whether a vehicle is accelerating or braking.

Try this

Q1. A particle has velocity $v = 5\ \text{m s}^{-1}$. State its direction of motion and its speed. [2 marks]

• Cue. Positive direction; speed $5\ \text{m s}^{-1}$.

Q2. A particle has $v = 3\ \text{m s}^{-1}$ and $a = -1\ \text{m s}^{-2}$. Is it speeding up or slowing down? [1 mark]

• Cue. Opposite signs, so it is slowing down.

Q3. Explain why a particle can have zero displacement yet have travelled a positive distance. [2 marks]

• Cue. It moved away from the origin and returned, so its net position change is zero but the path length is not.