60-second revision
- Speed = distance ÷ time; km/h × 5/18 = m/s and m/s × 18/5 = km/h.
- Same distance: time is inversely proportional to speed (speed 2:3 means time 3:2).
- Equal distances at x and y: average speed = 2xy/(x + y), always below the simple average.
- Equal times at x and y: average speed = (x + y)/2.
- Opposite directions: add speeds; same direction: subtract them.
- Late at s1, early at s2: distance = s1 × s2 × (total time gap) ÷ (s2 − s1).
- "A beats B by x m or t s" means B covers x metres in t seconds.
One formula, three forms
Speed = Distance ÷ Time, so Distance = Speed × Time and Time = Distance ÷ Speed. Every question in this chapter is this one relation, used with care over units and with a sensible choice of what stays constant.
| Convert | Multiply by | Example |
|---|---|---|
| km/h to m/s | 5/18 | 54 km/h = 54 × 5/18 = 15 m/s |
| m/s to km/h | 18/5 | 25 m/s = 25 × 18/5 = 90 km/h |
| minutes to hours | 1/60 | 45 minutes = 0.75 hour |
| hours to seconds | 3600 | 0.5 hour = 1800 seconds |
Proportion saves time
- Same distance: speed and time are inversely proportional. Speeds 5:6 mean times 6:5.
- Same time: distance is proportional to speed.
- Same speed: distance is proportional to time.
Worked example: walking at 5/6 of his usual speed, a man arrives 10 minutes late. New time = 6/5 of the usual time, so the extra 1/5 of the usual time = 10 minutes. Usual time = 50 minutes.
Average speed
Average speed is always total distance ÷ total time. You may take the plain average of two speeds only when the times spent at each speed are equal.
| Situation | Average speed | Example |
|---|---|---|
| Equal distances at x and y | 2xy/(x + y) | 30 and 50 km/h: 3000/80 = 37.5 km/h |
| Equal times at x and y | (x + y)/2 | 1 hour each at 30 and 50: 40 km/h |
| Three equal distances at x, y, z | 3xyz/(xy + yz + zx) | 20, 30, 60 km/h: 108000/3600 = 30 km/h |
| Anything else | Total distance ÷ total time | 100 km in 2 h, then 60 km in 2 h: 160/4 = 40 km/h |
Relative speed: meeting and chasing
- Moving towards each other: relative speed = sum of speeds. Meeting time = gap ÷ (s1 + s2).
- Moving in the same direction: relative speed = difference of speeds. Catch-up time = head start ÷ (s1 − s2).
Worked example: two walkers 27 km apart start towards each other at 4 km/h and 5 km/h. They meet after 27 ÷ 9 = 3 hours; the first has walked 12 km.
Worked example: a bus leaves at 9 am at 40 km/h; a car follows on the same road at 10 am at 60 km/h. Head start = 40 km, closing speed = 20 km/h, so the car catches the bus 2 hours later, at 12 noon, 120 km from the start.
Late and early
If a person is late at one speed and early at another, the gap between the two travel times is late minutes + early minutes. Then distance = s1 × s2 × gap (in hours) ÷ (s2 − s1).
Worked example: at 3 km/h a student is 10 minutes late; at 4 km/h he is 5 minutes early. Gap = 15 minutes = 1/4 hour. Distance = 3 × 4 × 1/4 ÷ 1 = 3 km. Check: 3 km takes 60 minutes at 3 km/h and 45 minutes at 4 km/h, a 15-minute gap.
Races
"A beats B by x metres or t seconds" means that when A finishes, B still needs t seconds to cover the last x metres. So B's speed = x/t, B's time = race length ÷ B's speed, and A's time = B's time − t.
Worked example: in a 400 m race A beats B by 20 m or 4 seconds. B's speed = 5 m/s, so B takes 80 s and A takes 76 s.
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