# Acceleration

> Speeding up, slowing down and turning. How quickly velocity changes, and the equations of motion.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/acceleration/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The traffic light test
Mia's mum and the car next to them stop at a red light. The light turns green. The other car, a small electric hatchback, shoots ahead. Mia's mum's old ute rolls forward slowly.

Ten seconds later both cars are doing 60 km/h. They reached the **same** speed. But the electric car got there much sooner at the start.

"It has better acceleration," says Mia's mum. She means it changed its speed **faster**.

**In short:** Acceleration is how quickly velocity changes: $a = \dfrac{v - u}{t}$, measured in m/s². Speeding up, slowing down and changing direction are all acceleration, and near Earth a falling object accelerates at about 9.8 m/s².

## What is acceleration?

**Acceleration** is how quickly velocity changes. It tells you how much the velocity goes up (or down) **every second**.

$$a = \frac{\text{change in velocity}}{\text{time taken}} = \frac{v - u}{t}$$

- $u$ = starting (initial) velocity
- $v$ = final velocity
- $t$ = time taken
- $a$ = acceleration

**Unit**: metres per second per second, written **m/s²** (or m s⁻²).

An acceleration of 3 m/s² means the velocity increases by 3 m/s every second.

### Worked example: A car pulls away
A car goes from rest (0 m/s) to 24 m/s in 8 s.

$$a = \frac{24 - 0}{8} = 3 \text{ m/s}^2$$

After 1 s it does 3 m/s, after 2 s it does 6 m/s, and so on.

## Slowing down

When an object slows down, $v$ is smaller than $u$, so the acceleration comes out **negative**. This is often called **deceleration**.

### Worked example: Braking bike
A bike slows from 10 m/s to 4 m/s in 3 s.

$$a = \frac{4 - 10}{3} = -2 \text{ m/s}^2$$

The bike decelerates at 2 m/s².

## Acceleration includes changing direction

Because velocity is a vector, changing **direction** also counts as acceleration, even if speed stays the same. A car going round a bend at a steady speed is accelerating towards the centre of the bend.

## Acceleration due to gravity

Near Earth's surface, anything falling freely (with no air resistance) speeds up by about **9.8 m/s every second**. We call this $g$.

$$g \approx 9.8 \text{ m/s}^2$$

In some exams you may be told to use $g = 10 \text{ m/s}^2$ to make the maths easier. Read the question.

### Key idea: Feather and hammer
On the Moon in 1971, astronaut David Scott dropped a hammer and a feather at the same time. With no air to slow the feather, both hit the ground together. Without air resistance, **all objects fall with the same acceleration**, whatever their mass.

## Using the formula three ways

$$a = \frac{v - u}{t} \qquad v = u + at \qquad t = \frac{v-u}{a}$$

### Worked example: How fast after falling?
A stone is dropped from rest. How fast is it moving after 3 s? (Use $g = 9.8 \text{ m/s}^2$.)

$$v = u + at = 0 + 9.8 \times 3 = 29.4 \text{ m/s}$$

**Mind map**

```text
Acceleration
  Meaning
    Change in velocity each second
    Speeding up: positive
    Slowing down: negative
    Turning also counts
  Formula
    a = (v − u) / t
    v = u + at
    Unit: m/s²
  Gravity
    g = 9.8 m/s²
    Same for all masses
    No air resistance
  Equations of motion
    Year 11
    s = ut + ½at²
    v² = u² + 2as
```

### Year 11: The equations of motion (SUVAT)
For **constant** acceleration in a straight line, five quantities are linked: $s$ (displacement), $u$, $v$, $a$, $t$. Every equation leaves one out.

$$v = u + at$$
$$s = ut + \tfrac{1}{2}at^2$$
$$v^2 = u^2 + 2as$$
$$s = \tfrac{1}{2}(u + v)t$$

**Method**: write down which three you know and which one you want. Pick the equation that contains those four.

**Example.** A car at 20 m/s brakes at $-5 \text{ m/s}^2$. How far does it travel before stopping?

Know: $u = 20$, $v = 0$, $a = -5$. Want: $s$. Use $v^2 = u^2 + 2as$:

$$0 = 400 + 2(-5)s \;\Rightarrow\; s = 40 \text{ m}$$

Notice: doubling the speed makes the braking distance **four times** longer, because it depends on $u^2$. That is why speed limits near schools are so low.

### Year 12: Centripetal acceleration
An object moving in a circle of radius $r$ at constant speed $v$ is always accelerating towards the centre:

$$a_c = \frac{v^2}{r}$$

The force causing this is the **centripetal force**, $F_c = \dfrac{mv^2}{r}$. It is not a new kind of force. It is whatever force is pulling inwards: friction for a car on a bend, tension for a ball on a string, gravity for a satellite.

## Quick summary

- Acceleration is the rate of change of velocity: $a = \dfrac{v-u}{t}$.
- Unit: m/s². A value of 2 m/s² means velocity changes by 2 m/s each second.
- Negative acceleration means slowing down (if moving in the positive direction).
- Changing direction is also acceleration.
- Free fall near Earth: $g \approx 9.8 \text{ m/s}^2$, the same for every mass if there is no air resistance.
- $v = u + at$ finds the final velocity.

## Quiz

Try these questions from the chapter quiz.

1. A scooter goes from 0 to 12 m/s in 4 s. Its acceleration is:
   - A. 48 m/s²
   - B. 3 m/s²
   - C. 0.33 m/s²
   - D. 16 m/s²
   - Answer: B. 3 m/s². a = (12 − 0) ÷ 4 = 3 m/s².

2. The unit of acceleration is:
   - A. m/s
   - B. m
   - C. m/s²
   - D. s/m
   - Answer: C. m/s². Acceleration is change in velocity (m/s) per second, so m/s².

3. A car slows from 20 m/s to 5 m/s in 5 s. Its acceleration is:
   - A. 3 m/s²
   - B. −3 m/s²
   - C. 5 m/s²
   - D. −5 m/s²
   - Answer: B. −3 m/s². a = (5 − 20) ÷ 5 = −3 m/s². The minus sign shows it is slowing down.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/acceleration/#quiz