Mia's family drives from Valencia to Barcelona. She records the distance shown on the GPS at regular intervals:
Time elapsed (h)
Distance from Valencia (km)
0
0
0.5
54
1.0
108
1.5
162
2.0
216
2.5
270
3.0
324
Before reading on: What do you notice about the numbers in the right column? What is happening every 0.5 hours? Can you calculate the car's speed from this table?
📐Maths connection: Look at the table again. Each row adds 54 to the distance and 0.5 to the time. The rate of change is constant: 54 ÷ 0.5 = 108 km/h. That's exactly the slope formula from maths lessons 16–18: slope = rise ÷ run = Δdistance ÷ Δtime.
💡 Deine Vermutung
If you plotted this data on a graph with time on the x-axis and distance on the y-axis, what would the graph look like? Sketch it quickly before reading further. What does the steepness of the line tell you?
📋 Die Grundformel: v = s / t
For an object moving at constant speed (gleichförmige Bewegung):
$$v = \frac{s}{t} \qquad s = v \cdot t \qquad t = \frac{s}{v}$$
Where: v = speed (Geschwindigkeit) · s = distance (Strecke/Weg) · t = time (Zeit)
[ s ]
[v] [t]
Cover the variable you want → what remains tells you how to calculate it.
Common units:
Speed: km/h (Stundenkilometer) or m/s (Meter pro Sekunde)
To convert: 1 km/h = 1 000 ÷ 3 600 m/s ≈ 0.278 m/s
Or: multiply km/h by 1 000/3 600 = 5/18 to get m/s
📈 Das Weg-Zeit-Diagramm (Distance-Time Graph)
Steeper slope = higher speed
Flat line (zero slope) = object is stationary
Negative slope = object moving back toward the start
The slope of the line equals the speed — this is exactly y = mx from maths
🧪 Ausprobieren — Measure your own speed
You need: a phone timer, a tape measure or a known distance (e.g. 10 floor tiles).
Measure out 10 metres (or estimate it using a known object).
Walk the distance and time yourself. Record: distance = 10 m, time = ? s
Calculate your walking speed in m/s using v = s ÷ t.
Convert to km/h (multiply by 3.6). Is this a normal walking speed? (Typical: 1.2–1.6 m/s)
Now run the same distance. How does your speed compare?
Worked Examples
Example 1 — Find the speed
Mia's train travels from Madrid to Málaga: 513 km in 2 hours 30 minutes. What is its average speed in km/h?
Convert time: 2 h 30 min = 2.5 hours
v = s ÷ t: 513 ÷ 2.5 = 205.2 km/h
This is the AVE high-speed train. At this speed the train covers about 57 m every second.
Example 2 — Two-segment journey
Mia cycles 12 km to a market at 15 km/h, then walks back 12 km at 5 km/h. What is her average speed for the whole trip?
Time cycling: t = s ÷ v = 12 ÷ 15 = 0.8 h
Time walking: t = 12 ÷ 5 = 2.4 h
Total distance: 12 + 12 = 24 km
Total time: 0.8 + 2.4 = 3.2 h
Average speed: 24 ÷ 3.2 = 7.5 km/h
⚠ Common mistake: (15 + 5) ÷ 2 = 10 km/h. This is wrong — you can't average speeds without accounting for the different times spent at each speed.
Problems
Problem 1
Complete the table. Use v = s/t, s = vt, or t = s/v.
Speed (km/h)
Distance (km)
Time (h)
80
240
?
120
?
3.5
?
450
5
65
?
2.25
?
12
0.4
— work on paper —
Problem 2
Mia walks 400 m to the beach in 6 minutes. What is her speed in (a) m/min, (b) m/s, and (c) km/h?
— work on paper —
Problem 3
A distance-time graph shows a straight line starting at (0, 0) and passing through (4 h, 300 km).
What is the slope of the line?
What does this slope represent physically?
What distance would be covered in 7 hours at the same speed?
— work on paper —
Problem 4
Bella the horse can canter at 20 km/h and gallop at 48 km/h. Mia canters for 15 minutes then gallops for 5 minutes. What total distance did they cover? Give your answer in km and in metres.
— work on paper —
Problem 5
Light travels at 299 792 km/s. The Moon is approximately 384 400 km from Earth. How long does it take light to travel from Earth to the Moon? Give your answer in seconds, and then in a more natural unit (minutes and seconds).
— work on paper —
Problem 6
Mia and her brother start at the same point. Mia walks at 4 km/h; her brother cycles at 18 km/h in the same direction. After how many minutes is her brother exactly 3.5 km ahead of her? Hint: think about relative speed.
— work on paper —
Problem 7 Challenge
A plane flies 2 400 km from London to Marrakech. Flying into a headwind, the journey takes 4 h. Flying back with a tailwind, it takes 3 h. The headwind and tailwind are the same speed w km/h. Set up two equations and find: (a) the plane's speed in still air, and (b) the wind speed.
— work on paper —
Problem 8 Open
Find the distance between two cities your family has visited (use Google Maps, set it to driving). Look up the typical driving speed limit for that route. Calculate: (a) expected travel time, (b) how the time changes if you drove 20% faster. Does 20% faster speed actually save 20% of the time?
— your research —
Science check ✔ — Average speed hides a lot. A car that drives 100 km in 1 hour might have stopped at traffic lights, driven at 130 km/h on the motorway, and crawled at 20 km/h through a town. The formula v = s/t gives the average over the whole journey — not what the speedometer showed at any single moment. In maths, instantaneous speed is the slope of a curve at a single point — which is the idea behind calculus.
Show Answer Key
1.3 h · 420 km · 90 km/h · 146.25 km · 30 km/h2.(a) 400 ÷ 6 ≈ 66.7 m/min · (b) 66.7 ÷ 60 ≈ 1.11 m/s · (c) 1.11 × 3.6 = 4 km/h3.(a) slope = 300 ÷ 4 = 75 km/h · (b) the speed of the vehicle · (c) 75 × 7 = 525 km4.Canter: 20 × (15/60) = 5 km · Gallop: 48 × (5/60) = 4 km · Total: 9 km = 9 000 m5.t = 384 400 ÷ 299 792 ≈ 1.282 s → about 1.28 seconds (just over a second)6.Relative speed = 18 − 4 = 14 km/h · t = 3.5 ÷ 14 = 0.25 h = 15 minutes7.Into headwind: (p − w) × 4 = 2 400 → p − w = 600. With tailwind: (p + w) × 3 = 2 400 → p + w = 800. Adding: 2p = 1 400 → p = 700 km/h. Then w = 800 − 700 = 100 km/h.8.Open — any correct calculation. Note: 20% faster speed gives 1/1.2 ≈ 83.3% of the original time, saving about 16.7% — not 20%. Speed and time are inversely related.
Next up → s04: Forces & Newton's Laws
What actually makes things speed up, slow down, or change direction? Newton figured out three rules that govern all motion — from horses to rockets.