Trains, Races & Circular Tracks

At a glance 11 formulas

1. 1Train past a platform$T = \dfrac{L_{\text{train}} + L_{\text{platform}}}{S}$$\text{pole or standing man: } T = \dfrac{L_{\text{train}}}{S}$
2. 2Two trains crossing$T = \dfrac{L_1 + L_2}{S_1 + S_2}\ \text{(opposite)}$$\text{same way: } T = \dfrac{L_1 + L_2}{S_1 - S_2}$
3. 3Passing a moving personA train passing a walker, a cyclist or a passenger in another train covers only its own length, at the relative speed: subtract if they move the same way, add if they meet.
4. 4Beats by x metres$\dfrac{S_A}{S_B} = \dfrac{L}{L - x}$
5. 5Margin in seconds$A \text{ beats } B \text{ by } x \text{ m or } t \text{ s} \Rightarrow S_B = \dfrac{x}{t}$$\text{winner's time} = \dfrac{L}{S_B} - t$
6. 6Chain the marginsIf A beats B by x m and B beats C by y m in an L m race, C has run $\frac{(L - x)(L - y)}{L}$ m when A finishes.
7. 7Head startA start of h m that ends in a dead heat is the same as beating by h m: in the faster runner's time the other runs $L - h$. A start of t seconds is t seconds of running alone.
8. 8First meeting anywhere$T = \dfrac{C}{a + b}\ \text{(opposite)}$$\text{same way: } T = \dfrac{C}{a - b}$
9. 9Together at the start$T = \text{LCM}\left(\dfrac{C}{a},\ \dfrac{C}{b}\right)$$\text{three runners: LCM of all three lap times}$
10. 10Distinct meeting points$a : b = p : q\ \text{in lowest terms}$$p + q\ \text{points opposite},\ p - q\ \text{the same way}$
11. 11Counting meetingsOpposite ways, they meet once for every lap their distances add up to; the same way, once for every lap the faster one gains. Count whole laps only.