# Time, Distance and Speed

> How fast is fast? Measuring motion with distance, time and speed, and changing units.

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

### Story time: The school run race
Every morning Leo rides his bike 3 km to school. His sister Ava walks a shorter path of 1.2 km. One day they leave home at the same moment. Leo arrives in 10 minutes. Ava arrives in 15 minutes.

"I'm faster," says Leo. "Your path was longer," says Ava. "That's not fair."

Who is really faster? You cannot tell just from the times, and you cannot tell just from the distances. You need both together. That combination is called **speed**.

**In short:** Speed is how much distance something covers each second: speed = distance ÷ time, or $v = \dfrac{d}{t}$. It is measured in metres per second (m/s) or kilometres per hour (km/h), and you divide km/h by 3.6 to get m/s.

## Three things that describe motion

- **Distance** is how far something travels. Unit: metre (m). We also use km and cm.
- **Time** is how long the motion takes. Unit: second (s). We also use minutes and hours.
- **Speed** is how much distance is covered **every second** (or every hour). Unit: metres per second (m/s) or kilometres per hour (km/h).

## The speed formula

$$\text{speed} = \frac{\text{distance}}{\text{time}} \qquad v = \frac{d}{t}$$

Physicists use $v$ for speed (from "velocity"), $d$ for distance and $t$ for time.

You can rearrange it in two other ways:

$$d = v \times t \qquad\qquad t = \frac{d}{v}$$

### Key idea: The triangle trick
Draw a triangle with **d** on top and **v** and **t** side by side underneath. Cover the one you want. What is left shows you the formula. Cover $v$: you see $d$ over $t$, so $v = d \div t$. Cover $d$: you see $v$ next to $t$, so $d = v \times t$.

### Worked example: Back to Leo and Ava
Leo: $v = \dfrac{3000 \text{ m}}{600 \text{ s}} = 5 \text{ m/s}$

Ava: $v = \dfrac{1200 \text{ m}}{900 \text{ s}} \approx 1.3 \text{ m/s}$

Leo really is faster, almost four times faster. Notice we changed km to m and minutes to seconds first.

## Changing units

Always check units before you put numbers into a formula.

- 1 km = 1000 m
- 1 hour = 60 minutes = 3600 s
- To change **km/h to m/s**, divide by 3.6
- To change **m/s to km/h**, multiply by 3.6

Why 3.6? Because $1 \text{ km/h} = \dfrac{1000 \text{ m}}{3600 \text{ s}} = \dfrac{1}{3.6} \text{ m/s}$.

### Worked example: A car on the highway
A car travels at 108 km/h. What is that in m/s?

$108 \div 3.6 = 30 \text{ m/s}$. Every second, the car moves 30 metres. That is about the length of three buses.

## Average speed and instantaneous speed

Leo does not ride at exactly 5 m/s the whole way. He slows at corners and speeds up on hills. So 5 m/s is his **average speed**.

$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

**Instantaneous speed** is the speed at one exact moment. A car's speedometer shows instantaneous speed. A speed camera that checks your time between two points measures average speed.

### Watch out: Do not average the speeds
A runner goes 100 m at 10 m/s and then 100 m at 5 m/s. The average speed is **not** 7.5 m/s.

Time for first part: $100 \div 10 = 10$ s. Second part: $100 \div 5 = 20$ s.

Average speed $= \dfrac{200}{30} \approx 6.7$ m/s. Always use total distance over total time.

## Measuring speed in the lab

- **Stopwatch and tape measure**: simple, but your reaction time adds error (about 0.2 s).
- **Light gates**: a beam is broken when the object passes. A computer records time very precisely.
- **Ticker timer**: puts dots on a paper tape at equal time gaps. Dots far apart mean fast. Dots close together mean slow.

**Mind map**

```text
Speed
  Formula
    v = d ÷ t
    d = v × t
    t = d ÷ v
  Units
    m/s
    km/h
    km/h ÷ 3.6 = m/s
  Types
    Average: total d ÷ total t
    Instantaneous: right now
  Measuring
    Stopwatch
    Light gates
    Ticker timer
```

### Year 11: Distance versus displacement
In Year 11 you separate **distance** (total path length, no direction) from **displacement** (straight line change in position, with direction). This leads to **speed** versus **velocity**.

If you run once around a 400 m track and stop where you started, your distance is 400 m but your displacement is **zero**. Your average speed is positive but your average velocity is zero.

$$\vec{v}_{\text{avg}} = \frac{\Delta \vec{s}}{\Delta t}$$

The next chapter, Vectors and Scalars, explains this fully.

### Year 12: Relative velocity
Speeds depend on who is watching. Two cars drive towards each other at 20 m/s and 25 m/s. To each driver, the other car approaches at $20 + 25 = 45$ m/s. If they drive the same way, one sees the other move at only $25 - 20 = 5$ m/s.

$$\vec{v}_{\text{A rel B}} = \vec{v}_A - \vec{v}_B$$

At very high speeds, close to the speed of light, this simple rule breaks down. Einstein's special relativity takes over, which you study in Year 12 Physics.

## Quick summary

- Speed tells you how much distance is covered per unit of time.
- $v = \dfrac{d}{t}$, $d = vt$, $t = \dfrac{d}{v}$.
- Change units first: km to m, minutes or hours to seconds.
- km/h $\div$ 3.6 = m/s. m/s $\times$ 3.6 = km/h.
- Average speed = total distance ÷ total time. Never just average the speeds.
- Instantaneous speed is the speed at one moment, like a speedometer reading.

## Quiz

Try these questions from the chapter quiz.

1. A cyclist travels 600 m in 120 s. What is the average speed?
   - A. 0.2 m/s
   - B. 5 m/s
   - C. 72 m/s
   - D. 720 m/s
   - Answer: B. 5 m/s. v = d ÷ t = 600 ÷ 120 = 5 m/s.

2. A train moves at 25 m/s for 40 s. How far does it go?
   - A. 0.625 m
   - B. 65 m
   - C. 1000 m
   - D. 1600 m
   - Answer: C. 1000 m. d = v × t = 25 × 40 = 1000 m.

3. How long does it take to walk 450 m at 1.5 m/s?
   - A. 675 s
   - B. 300 s
   - C. 30 s
   - D. 0.0033 s
   - Answer: B. 300 s. t = d ÷ v = 450 ÷ 1.5 = 300 s, which is 5 minutes.

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