How fast? Distance, time and speed
By the end of this lesson you will be able to
- Define distance, time and speed, and state the SI unit of each.
- Calculate speed, distance or time from the other two using speed = distance ÷ time, with working and units.
- Convert between m/s and km/h and choose the right unit for a situation.
- Distinguish average speed from instantaneous speed, and say which one a given instrument measures.
- Describe a method for measuring a speed accurately, naming the instruments and the main source of error.
The puzzle
In Paris in August 2024, Letsile Tebogo of Botswana ran 200 m in 19.46 s and became Olympic champion. A cheetah in the Kalahari can run at 30 m/s. A car on the A1 to Francistown does 120 km/h. Which of the three is fastest? You cannot tell until all three are written in the same language — and that language is the subject of this lesson.
Before reading on, write down your guess: which is fastest, and by roughly how much? Then write one sentence explaining what “fast” actually means — what two things do you need to know to compare speeds?
Compare your answer
To compare speeds you need how far something moved and how long it took — a distance and a time. Tebogo: 200 m ÷ 19.46 s ≈ 10.3 m/s. The car: 120 km/h = 120 000 m ÷ 3600 s ≈ 33.3 m/s. The cheetah: 30 m/s. So the car is fastest, the cheetah is just behind it, and the fastest human on Earth over 200 m is running at less than a third of the car’s speed. Most people guess the cheetah beats the car — it does not, though it would win the first 50 m.
The science — structured notes
1.1 Describing movement: distance and time
Something is moving when its position changes with time. To describe a movement scientifically you need two measurements, and both must have units:
- Distance — how far the object has travelled along its path. SI unit: the metre (m). Also used: centimetres (cm) for a bench experiment, kilometres (km) for a road trip.
- Time — how long the movement took. SI unit: the second (s). Minutes and hours are convenient but must be converted before any calculation in m/s.
Distance is measured along the actual path. If you walk 300 m to the shop and 300 m back, the distance you travelled is 600 m even though you finished exactly where you started. (Physicists have a separate word, displacement, for “how far from where you started, and in which direction” — you will meet it in Year 4. In this unit, “distance” always means the path length.)
A football referee runs to the halfway line (50 m), back to the goal line (50 m), then to the centre spot (50 m). What distance has she run?
Add the lengths of each part of the path: 50 + 50 + 50 = 150 m. Where she ends up (the centre spot) does not matter for distance — only the path does.
A combi drives 12 km from Mogoditshane to the Gaborone bus rank, then 12 km back.
(a) State the distance travelled.
(b) State how far the combi is from where it started.
(c) Explain why a passenger reading a book feels as if she is not moving.
Reveal answer
(a) 12 + 12 = 24 km — distance is the length of the whole path.
(b) 0 km — it is back where it started.
(c) Relative to the combi (her seat, the book, the other passengers) her position is not changing, so relative to the combi she is at rest. Relative to the road she is moving at the combi’s speed. Both statements are true; they use different reference points.
1.2 Speed: distance covered per unit of time
Speed is the distance travelled in each unit of time — how many metres every second, or how many kilometres every hour. The definition is an equation:
speed = distance ÷ time or v = dt
Units: metres per second (m/s) when distance is in m and time in s; kilometres per hour (km/h) when distance is in km and time in h.
Rearranged: distance = speed × time and time = distance ÷ speed.
Tebogo ran 200 m in 19.46 s. Calculate his average speed.
Step 1 — Write the equation: speed = distance ÷ time.
Step 2 — Substitute with units: speed = 200 m ÷ 19.46 s.
Step 3 — Calculate: = 10.28 m/s. Round to a sensible number of figures: 10.3 m/s.
Three lines. The examiner gives a mark for the equation or substitution, a mark for the number, and requires the unit. A bare “10.3” can lose the last mark.
A cheetah runs at 30 m/s for 8 s. How far does it run? How long would it take to cover 450 m at that speed?
distance = speed × time = 30 m/s × 8 s = 240 m.
time = distance ÷ speed = 450 m ÷ 30 m/s = 15 s. (A real cheetah cannot keep 30 m/s for 15 s — it overheats after about 20 s of sprinting — which is why a chase either ends quickly or fails.)
Show your working and give units.
(a) A snail moves 45 cm in 5 minutes. Calculate its speed in cm/s.
(b) Sound travels at 340 m/s. How far away is a lightning strike if the thunder arrives 6 s after the flash?
(c) A bus travels the 200 km from Gaborone to Mahalapye at an average speed of 80 km/h. How long does the journey take?
Reveal answer
(a) 5 min = 300 s. speed = 45 cm ÷ 300 s = 0.15 cm/s.
(b) distance = speed × time = 340 m/s × 6 s = 2040 m ≈ 2 km. (The light arrives almost instantly — that is why you count from the flash.)
(c) time = distance ÷ speed = 200 km ÷ 80 km/h = 2.5 h = 2 h 30 min. Notice that the km cancel and h is left — the units tell you the answer is a time.
Common slip in (a): dividing by 5 instead of 300. Convert the time to seconds first if the answer must be in cm/s.
1.3 Units: m/s and km/h, and how to convert
Both units are correct; which one you use depends on the situation. A running track is measured in metres and seconds, so m/s. A road is measured in kilometres and hours, so km/h. Converting between them is one of the most common exam tasks, and it is easiest if you go back to the definition:
Convert 120 km/h to m/s.
Step 1 — 120 km/h means 120 km in 1 hour.
Step 2 — Convert each unit: 120 km = 120 000 m; 1 h = 3600 s.
Step 3 — Divide: 120 000 m ÷ 3600 s = 33.3 m/s.
Shortcut: km/h ÷ 3.6 = m/s, and m/s × 3.6 = km/h. The 3.6 comes from 3600 ÷ 1000. Learn where it comes from, not just the number, and you can never get it backwards.
| Mover | Typical speed / m/s | Typical speed / km/h |
|---|---|---|
| Walking | 1.4 | 5 |
| Olympic 200 m sprinter (average) | 10.3 | 37 |
| Cheetah (top speed) | 30 | 108 |
| Car on the highway | 33 | 120 |
| Sound in air | 340 | 1 224 |
| Passenger jet | 250 | 900 |
| Light | 300 000 000 | 1 080 000 000 |
A student writes: “50 km/h = 50 × 1000 = 50 000 m/s.” What went wrong, and what is the correct answer?
Reveal answer
The student converted the kilometres to metres but not the hours to seconds. 50 km/h = 50 000 m ÷ 3600 s = 13.9 m/s (or 50 ÷ 3.6). A sanity check would have caught it: 50 000 m/s is 50 km every second — faster than any rocket.
1.4 Average speed and instantaneous speed
Tebogo’s 10.3 m/s is an average speed: total distance ÷ total time. It hides the detail. For the first second he was almost stationary; around 60 m into the race he was probably running at over 11.5 m/s; in the last 30 m he was slowing. The speed at one instant — what a speedometer or a speed camera shows — is the instantaneous speed.
| Average speed | Instantaneous speed | |
|---|---|---|
| Definition | Total distance ÷ total time for a whole journey | Speed at one moment |
| Measured by | Tape/trundle wheel + stopwatch; trip meter + clock | Speedometer; radar gun; light gates very close together |
| Example | Bus: 200 km in 2.5 h → 80 km/h, though it stopped at Dibete | Speed camera reads 128 km/h at that spot |
A cyclist rides 6 km in 15 min, rests for 5 min, then rides 4 km in 10 min. Find her average speed for the whole trip in km/h.
Total distance = 6 + 4 = 10 km. Total time = 15 + 5 + 10 = 30 min = 0.5 h (the rest counts!).
Average speed = 10 km ÷ 0.5 h = 20 km/h. Not the average of 24 km/h and 24 km/h for the two riding parts — the stop pulls it down.
“A car with an average speed of 60 km/h over a journey must have been travelling at 60 km/h at some moment during it.”
Reveal answer
True — but for a subtle reason. If the car was ever below 60 km/h it must also have been above 60 km/h to average out (otherwise the average would be below 60), and since speed changes smoothly it must have passed through exactly 60 km/h on the way. The point of the question: an average tells you the speed was 60 km/h at least once, but tells you nothing about how long it stayed there.
1.5 Measuring speed well
Speed is never measured directly. You measure a distance and a time and divide. So the quality of a speed measurement depends on two instruments and the way they are used:
“Mark a 10.0 m course on the flat playground with a trundle wheel. The runner starts 5 m before the first mark so that she crosses it at full speed. Two students with stopwatches start when her chest crosses the first line and stop at the second; the mean of the two is used. Repeat three times and calculate the mean time; speed = 10.0 m ÷ mean time. Reaction time (about 0.2 s) is the main source of error; light gates at the two lines would remove it.”
Try it — the speed trap
A battery car runs through two light gates. Set the distance between the gates five times (20, 40, 60, 80, 100 cm), three trials each, and record the time. Then add a column by hand: distance ÷ time for every row. If the car really moves at a steady speed, every answer will be the same number — and that number is the speed. In Results & graph, plot time against distance and describe the line: is it straight? through the origin? What is the gradient, and what does 1 ÷ gradient give you?
Suppose your mean times were: 20 cm → 0.44 s, 40 cm → 0.89 s, 60 cm → 1.33 s, 80 cm → 1.78 s, 100 cm → 2.22 s.
(a) Calculate the speed from the 100 cm row in cm/s and in m/s.
(b) Is time directly proportional to distance? Give the evidence.
(c) Predict the time for 150 cm and say how much you trust the prediction.
Reveal answer
(a) 100 cm ÷ 2.22 s = 45.0 cm/s = 0.450 m/s.
(b) Yes. Doubling the distance (20 → 40 cm) doubles the time (0.44 → 0.89 s, within error); the ratio time ÷ distance is 0.022 s/cm for every row; the graph is a straight line through the origin.
(c) time = 150 cm ÷ 45 cm/s = 3.3 s. This is extrapolation beyond the measured range — the car might slow as the battery drains over a longer run — so it is a reasonable estimate but should be tested.
Common mistakes to avoid
- No units, or the wrong units — “10.3” is not a speed; “10.3 m/s” is. Write the unit at every line of working.
- Converting only half the unit — km/h to m/s needs both km → m (×1000) and h → s (÷3600).
- Forgetting to convert minutes to seconds before dividing when the answer must be in m/s.
- Averaging speeds instead of using total distance ÷ total time — stops count in the time.
- Calling the average speed “the speed the whole way” — it is a summary of the journey.
- A method with “push the car” and “time it” — say how the speed is set, how the distance is marked and how the timer is triggered.
How we found out — the speed of light was measured with a moon
For two thousand years philosophers argued whether light travels instantly or merely very fast. Galileo tried to measure it in 1638 with two lanterns on hilltops a mile apart: uncover one, and the assistant uncovers his the moment he sees the flash. All he measured was human reaction time. Light was too quick for any clock on Earth.
In 1676 a Danish astronomer in Paris, Ole Rømer, was timing the eclipses of Io, a moon of Jupiter, which disappears behind the planet like clockwork every 42.5 hours. He noticed that the eclipses ran up to 22 minutes late when Earth was on the far side of its orbit from Jupiter, and early when it was on the near side. His explanation was speed = distance ÷ time in reverse: the light from Io had to cross the extra width of Earth’s orbit, and that took time. From the delay and the size of the orbit he estimated that light travels about 220 000 km every second — within a third of the modern value of 299 792 km/s, using a moon as his light source and the Solar System as his running track.
The lesson is the one this unit keeps returning to: a speed is always a distance and a time. Rømer could not use a stopwatch, so he found a clock in the sky and a distance in the heavens, and the equation did the rest.
Why did Galileo’s lantern experiment fail, and why did Rømer’s succeed, when both used the same equation?
Reveal answer
Both needed distance ÷ time. Galileo’s distance (about 1.6 km) gave a light travel time of 0.000005 s — far smaller than the reaction-time error of about 0.2 s, so the true value was buried in error. Rømer’s distance was the diameter of Earth’s orbit (about 300 million km), giving a delay of many minutes, which even a 17th-century clock could measure. The equation was the same; Rømer chose a distance large enough to make the time measurable. Choosing a measurement big enough to beat your error is a design skill you will use in every investigation.
Where it is used now
- Botswana — Letsile Tebogo’s coaches break his 200 m into 50 m splits. His fastest split at Paris 2024 was under 4.5 s — over 11 m/s — and the numbers, not opinions, decide what to train next.
- Roads. Average-speed cameras on highways photograph a car at two points a known distance apart and calculate distance ÷ time; you cannot beat them by braking at the camera.
- Weather radar and lightning. The 3-seconds-per-kilometre rule for thunder is speed = distance ÷ time rearranged, with the speed of sound built in.
- Delivery and ride apps estimate arrival by dividing the remaining distance by the average speed of the last few minutes of traffic.
Science now — and the scientist you need to be
What is happening in 2026
Speed used to be measured with a stopwatch; now it is measured by satellites. A phone’s GPS receiver hears timing signals from at least four satellites and works out where it is to a few metres, thirty times a second — and every sports watch, ride-share app and delivery van turns those positions into speeds by the same equation you used today. In athletics, AI systems analyse video frame by frame to give every sprinter a speed curve for the whole race; the coaches at Paris 2024 had Tebogo’s split times before he had crossed the line. On the roads, cameras with number-plate recognition and average-speed sections are spreading across Southern Africa because the physics is simple and the evidence is hard to argue with.
What has not changed is that every one of these systems can be wrong, and only someone who understands what a speed is can tell. A GPS watch that loses satellites under trees invents a zig-zag path and reports a speed that never happened. An AI that mis-identifies which frame the runner’s chest crossed the line changes a split by a hundredth of a second. The machine does the dividing; the human must know what the distance and time actually were.
The mindshift
What you need to be able to do
Every technical job — engineering, medicine, logistics, sport science, data analysis — rests on rates: something per something. The skill this lesson builds is not the equation but the habit of asking “per what?”, carrying units through a calculation, and sanity-checking a result against something you know. An engineer who notices that a computed speed is faster than sound has caught a million-pula mistake. That habit is what employers and universities mean by numeracy, and no calculator or AI supplies it for you.
See an example
| Method | Distance / m | Trial 1 time / s | Trial 2 time / s | Trial 3 time / s | Mean time / s | Speed / m/s |
|---|---|---|---|---|---|---|
| Tape and stopwatch | 20.0 | 14.2 | 14.6 | 14.4 | 14.4 | 1.39 |
| Phone app | 20.0 | — | — | — | — | 1.6 |
Tape and stopwatch: 20.0 m in 14.2, 14.6, 14.4 s → mean 14.4 s → 1.39 m/s. Phone app: reported 1.6 m/s but the map showed the path wandering 3 m sideways, so it measured a longer distance than I walked. I trust the tape more: its distance is certain to ±2 cm; the stopwatch reaction time (±0.2 s of 14.4 s) is only about 1.5 %. To match light gates I would have a second person time from a fixed line and start me from 3 m behind the line so I cross it at a steady speed.
Key terms
| Distance | The length of the path an object has travelled; SI unit the metre (m). |
| Time | How long a movement takes; SI unit the second (s). |
| Speed | The distance travelled per unit time: speed = distance ÷ time; units m/s or km/h. |
| Average speed | Total distance ÷ total time for a whole journey, including stops. |
| Instantaneous speed | The speed at one particular moment, as shown by a speedometer. |
| Reference point | The thing an object’s movement is measured relative to — usually the ground unless stated. |
| Light gate | A beam-and-sensor timer that starts or stops a clock when an object breaks the beam, removing human reaction time. |
| Reaction time | The delay (about 0.2 s) between seeing an event and pressing a stopwatch; a source of random error. |
Think like a scientist
Criterion A The Gaborone–Francistown road is 430 km. A driver claims: “I did it in 3 hours flat, and I never went above the 120 km/h limit.”
(a) Calculate the driver’s average speed.
(b) Explain whether the claim about the speed limit can be true if the driver stopped for fuel for 15 minutes.
(c) An average-speed camera system records the car passing two points 20.0 km apart, 9 minutes 30 seconds apart. Calculate the average speed over that section in km/h and state what it shows.
Model answer
(a) speed = distance ÷ time = 430 km ÷ 3 h = 143 km/h (average).
(b) With a 15 min stop the driving time is 2.75 h, so the average while moving was 430 ÷ 2.75 = 156 km/h. An average of 156 km/h cannot be achieved without being above 156 km/h at some moments, let alone above 120 km/h. The claim cannot be true.
(c) 9 min 30 s = 9.5 min = 0.158 h. speed = 20.0 km ÷ 0.158 h = 126 km/h. Over that section the car averaged more than the limit, so at some points it must have exceeded 126 km/h — the camera provides evidence without needing to catch the instant.
Probe deeper — exam-style questions
Attempt every question before opening the answer. Marks in brackets show what a full answer is worth.
Define speed and state its SI unit.
Reveal answer
Speed is the distance travelled per unit time (distance ÷ time) [1]; SI unit metres per second, m/s [1].
A train travels 240 km in 1 h 36 min. Calculate its average speed in km/h and in m/s.
Reveal answer
1 h 36 min = 1.6 h [1]. 240 ÷ 1.6 = 150 km/h [1]. 150 ÷ 3.6 = 41.7 m/s [1].
A student measures the speed of a toy car by pushing it and timing it over 1 m with a stopwatch. Give three improvements that would make the measured speed more reliable, and justify each.
Reveal answer
Any three of: release the car from a fixed height on a ramp (or a fixed motor setting) so its speed is reproducible, rather than a hand push [1]; use light gates/electronic timer to remove reaction-time error [1]; use a longer distance so the time is large compared with the error [1]; repeat three times and take the mean, discarding anomalies [1]. Max 3.
“A car’s average speed on a trip was 80 km/h, so it was doing 80 km/h when it passed the school.” Explain the error.
Reveal answer
Average speed is total distance ÷ total time for the whole journey [1]; it says nothing about the instantaneous speed at any particular place, which could have been anything from 0 (stopped) to well above 80 km/h [1].
Light travels at 300 000 km/s. The Moon is 384 000 km away. Calculate how long light from the Moon takes to reach Earth, and explain what you are actually seeing when you look at the Moon.
Reveal answer
time = distance ÷ speed = 384 000 ÷ 300 000 = 1.28 s [2: equation/substitution, answer with unit]. You see the Moon as it was 1.28 s ago — all seeing is looking into the past, by a delay of distance ÷ speed of light [1].
A speed limit sign says 60 km/h. A learner says “that’s about 60 m/s, so about 60 metres every second”. Calculate the correct value in m/s and explain, using the size of the number, why the learner’s version would be dangerous to believe when judging whether it is safe to cross the road.
Reveal answer
60 ÷ 3.6 = 16.7 m/s [1]. The learner’s figure is 3.6 times too large [1]. A pedestrian who believes a car covers 60 m every second would think a car 100 m away arrives in under 2 s and never cross; more dangerously, a learner who made the opposite error (60 km/h as 6 m/s) would step out in front of a car that actually covers 17 m every second [1]. Judging a gap in traffic is a speed = distance ÷ time calculation done in the head — the unit matters.
Summary
- Movement is a change of position relative to a reference point; distance is the path length (m) and time is measured in seconds.
- speed = distance ÷ time; rearrange for distance (= speed × time) and time (= distance ÷ speed). Always show the equation, the substitution with units, and the answer with its unit.
- km/h ÷ 3.6 = m/s. Convert both parts of the unit, and sanity-check the size of the answer.
- Average speed is total distance ÷ total time (stops included); instantaneous speed is the speed at one moment.
- A good speed measurement fixes the distance, produces the speed reproducibly, triggers the timer without human reaction time (light gates) and repeats.
Next time → Distance–time graphs: a whole journey in one picture, and why the slope is the speed.