Part 2 · Trains, Boats, Races & Circular Tracks

Formula sheet

Part 2 of Time, Speed & Distance. Everything in part 1 still applies — this page is only what each application adds.

Trains — the moving thing has length


Crossing a pole, post or person standing still	distance $=$ the train's length
Crossing a platform, bridge or tunnel	distance $=$ train $+$ platform
Crossing a person who is walking	distance $=$ train's length, at the relative speed
Two trains crossing each other	distance $=$ sum of both lengths, at the relative speed
Overtaking, same direction	relative speed $= s_1 - s_2$
Passing, opposite directions	relative speed $= s_1 + s_2$

Boats and streams — the medium helps or hinders


Downstream	$b + r$
Upstream	$b - r$
Boat's own speed	$b = \dfrac{\text{down} + \text{up}}{2}$
Stream's speed	$r = \dfrac{\text{down} - \text{up}}{2}$
There and back, distance $d$ each way	$\dfrac{d}{b+r} + \dfrac{d}{b-r} = \dfrac{2bd}{b^2-r^2}$
The same maths governs	escalators, moving walkways, and a plane in wind

Races — everyone runs for the same length of time


"A beats B by $d$ metres"	when A has run $L$ , B has run $L - d$
"A beats B by $t$ seconds"	B needs $t$ more seconds for the same $L$
"A gives B a start of $d$ metres"	A runs $L$ while B runs $L-d$
So the speed ratio is	$\dfrac{s_A}{s_B} = \dfrac{L}{L-d}$
Chaining two results	multiply the ratios — this is how three-runner questions close

Circular tracks — the track repeats


Same direction, time between meetings	$\dfrac{L}{s_1 - s_2}$
Opposite directions, time between meetings	$\dfrac{L}{s_1 + s_2}$
First meeting at the starting point	$\operatorname{lcm}$ of the individual lap times
Distinct meeting points, speeds in lowest ratio $a:b$	$\lvert a-b \rvert$ same way, $a+b$ opposite ways