Lesson 2 of 2 | Totals, Weights and Units
Average Speed: Equal Distance Is Not Equal Time
Start with distance divided by time
Average speed is total distance divided by total elapsed time. Add distances and times separately before dividing. The simple arithmetic mean of two speeds is justified for equal time at each speed, not for equal distance. State whether stops count in the journey time.
Equal-distance example
A traveller covers 90 km at 45 km/h and returns 90 km at 30 km/h, without stopping. The outward leg takes 2 hours and the return takes 3 hours. The complete journey is 180 km in 5 hours: average speed = 36 km/h.
The tempting midpoint, 37.5 km/h, is wrong because the traveller spends more time at the slower speed. For two equal-distance legs with positive speeds u and v, the formula 2uv/(u+v) follows from 2d divided by (d/u + d/v). Deriving it is safer than applying it without checking the equal-distance condition.
Equal-time example
Now travel for two hours at 45 km/h and two hours at 30 km/h. Distances are 90 km and 60 km, totaling 150 km in 4 hours. Average speed is 37.5 km/h. The arithmetic mean works here because time weights are equal.
Include a stop when the question includes it
Return to the 180 km journey taking 5 hours of movement. If a 30-minute stop is included in total elapsed time, divide 180 by 5.5, giving approximately 32.73 km/h. Do not divide by 5.30: thirty minutes is half an hour, not 0.30 hours.
Speed is not velocity
A return journey can have nonzero average speed even though its net displacement is zero. Average velocity uses displacement, whereas average speed uses distance travelled. Identify the quantity requested before choosing the numerator.
A reliable scratch-work table
| Leg | Distance | Speed | Time |
|---|---|---|---|
| Outward | 90 km | 45 km/h | 2 h |
| Return | 90 km | 30 km/h | 3 h |
| Total | 180 km | Do not add speeds | 5 h |
Original practice
- Travel 80 km at 40 km/h and 120 km at 60 km/h with no stops. Find average speed.
- Cover two equal-distance legs at 20 and 30 km/h. Find average speed.
- Cover 150 km in 3 hours of movement plus a 45-minute stop. Find average speed including the stop.
Worked answers: 1. The legs take 2 hours each, so 200/4 = 50 km/h. 2. Choose 60 km per leg: 120/(3+2) = 24 km/h. 3. Elapsed time is 3.75 hours, so 150/3.75 = 40 km/h.
Check units, the meaning of total time and the plausibility of the result before choosing an option. These exercises teach arithmetic foundations and do not claim a particular future exam pattern.