What is Average speed?

Example 61.2

A train covers $270$ km in $1$ h $30$ min. First convert the time: $1$ h $30$ min $= 1.5$ h. Then

Distance at $90$ km/h during $2$ h $20$ min $= \frac73$ h: $d = 90 \times \frac73 = 210$ km. Time for $35$ km at $14$ km/h: $t = \frac{35}{14} = 2.5$ h $= 2$ h $30$ min.

Example 61.6 (Flow, density, price per kilo)

Speed has many cousins, all treated the same way:

• a tap fills $48$ L in $4$ min: flow rate $\frac{48}{4} = 12$ L/min, so filling a $150$ L tub takes $\frac{150}{12} = 12.5$ min;
• $0.6$ kg of cheese costs $9$ euros: price $\frac{9}{0.6} = 15$ euros per kg;
• a car uses $6.3$ L for $90$ km: consumption $\frac{6.3}{90} \times 100 = 7$ L per $100$ km.

Identify the quotient, and the three-way formula ($q = \frac ab$, $a = qb$, $b = \frac aq$) does the rest.

Example 61.9 (Average speed over two legs)

A cyclist rides $30$ km at $30$ km/h, then $30$ km at $15$ km/h. The average speed over the whole trip is not $\frac{30 + 15}{2} = 22.5$ km/h. Compute with the definition:

1. times: $\frac{30}{30} = 1$ h, then $\frac{30}{15} = 2$ h; total $3$ h;
2. total distance $60$ km, so $v = \frac{60}{3} = 20$ km/h.

The slow leg lasts longer, so it weighs more — an average of speeds must be weighted by time.