Section 32.2

Rates, Work and Motion

Combined work rates, relative speeds, average speed as a harmonic mean, the fly between trains, clock hands, couriers.

Toolkit

Model every moving or working agent by a constant rate (distance/time or work/time); rates combine additively, with opposing effects (current, drain, closing vs separating motion) entered with sign, and total time is total quantity divided by net rate.

Key results:

Combined work: if agent $i$ completes a job in $t_i$ , combined rate is $\sum 1/t_i$ , so joint time is $1/\sum(1/t_i)$ ; a drain or opposing agent contributes $-1/t_i$ .
With/against a current: effective speeds are $r+w$ and $r-w$ ; average speed over a round trip is the harmonic mean $\frac{2(r+w)(r-w)}{2r}$ , not $(v_1+v_2)/2$ .
Average speed over equal distances: $\bar v = \dfrac{2v_1v_2}{v_1+v_2}$ (harmonic mean); more generally $\bar v = \dfrac{\sum d_i}{\sum d_i/v_i}$ .
Relative speed: closing speed $=$ sum of speeds (opposite directions), difference of speeds (same direction); meeting/passing time $=$ distance/relative speed.
Chained ratios: if $A$ beats $B$ by factor $p$ and $B$ beats $C$ by factor $q$ over the same distance, $A$ beats $C$ by factor $pq$ .
Clock hands: minute hand moves $6^\circ$ /min, hour hand $0.5^\circ$ /min; angle between them at time $t$ (minutes past 12) is $|6t - (0.5t + \text{offset})|$ , reduced mod $360^\circ$ to $\le180^\circ$ . Hands coincide every $\dfrac{1}{1-\frac{1}{12}}=\dfrac{12}{11}$ hours.
Fly between approaching objects: total fly distance $=$ fly speed $\times$ (time until the two objects meet), avoiding summing an infinite series.
Geometric-mean meeting time: if ships meet after time $T$ and then take $t_1,t_2$ to reach the other’s start, $T=\sqrt{t_1 t_2}$ .

Problems