Speed, distance and time

Why speed is a rate (distance per one unit of time), how the three formulas are one relationship, why time must be in decimal hours, and why average speed is total distance over total time — never the mean of the speeds.

Ratio, proportion & rates of change · Compound units & rates

Speed, distance and time

Why it works

Speed is a rate: how much distance is covered in one unit of time. A speed of 60 km/h literally reads "60 kilometres per (each) hour". That "per" is a division, and it gives the defining formula

\text{speed} = \frac{\text{distance}}{\text{time}}.

The other two forms aren't separate facts to memorise — they are the same relationship rearranged. If each hour covers 60 km, then

t

hours cover

60t

km (

\text{distance} = \text{speed} \times \text{time}

), and covering 150 km takes

\frac{150}{60}

hours (

\text{time} = \frac{\text{distance}}{\text{speed}}

). Sense-check every rearrangement: distance should come out biggest, and a faster speed should make the time smaller.

Time must be in decimal hours — and minutes are not decimals. The single most common slip is writing 2 hours 30 minutes as

2.3

. Minutes are sixtieths of an hour, not hundredths:

2

30

min

= 2\frac{30}{60} = 2.5

hours. The same trap runs in reverse: an answer of

0.75

hours is

0.75 \times 60 = 45

minutes, not "75 minutes". Convert at the start, convert back at the end, and always through 60.

Average speed is total distance over total time. If a journey has two stages, the average speed is

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

not the mean of the two speeds. Drive 120 km at 60 km/h (2 hours) and then 120 km at 40 km/h (3 hours): that's 240 km in 5 hours, an average of 48 km/h — not 50. Why is it less than the mean? Because the journey spends more time at the slower speed, so the slow stage counts for more. The two speeds would only average to 50 if the times (not the distances) were equal.

Units must match before any formula is used. km with hours gives km/h; m with seconds gives m/s. Mixing them — kilometres with minutes, say — produces a number in no sensible unit at all. Convert everything first.