Otimization on a city with infinite many traffic lights.

Question

Province Ave has infinitely many traffic lights, equally spaced and synchronized. The distance between any two consecutive ones is $1500m$. The traffic light stay green for 1.5 minutes, red for 1 minute then green again for 1.5 minutes and so on. Suppose that a car is passing through Province Ave at constant speed equal to $\nu~m/s$, For which values for $\nu$ is it possible for the car pass through an arbitrarily large number of traffic light without stopping?

Answer 1

The green-red loop of the traffic lights has periodicity $1.5$minutes $+$ $1minute$ $=$ $150$seconds. It follows that crossing the first traffic light when it is green, the next one will surely be green after $150$seconds.

The time to pass from a traffic light to the next one at speed $\nu$ $m/s$ is $$ t = \frac{1500m}{\nu~\frac ms} = \frac{1500}{\nu} s $$ It then suffices to have $\nu$ such that $t=150$, i.e. $$ \frac{1500}{\nu} = 150 ~~\Rightarrow~~ \nu = 10 ~m/s $$

Answer 2

Speeds: $~~\nu ~=~ 20$ m/s$,$ $~ 10$ m/s$,$ $~ 6 \frac{2}{3}$ m/s$,$ $~ 5$ m/s$,$ $~ 3 \frac{1}{3}$ m/s$,$ $~ 2 \frac{1}{2}$ m/s$,\dots$
Kilometers per hour: 72, 36, 24, 18, 12, 9, . . .
Approximate miles per hour: 45, 22, 15, 11, 7, 6, . . .

These correspond to the fractions $~ 3000/150,$ $~ 1500/150,$ $~ 3000/450,$
              $1500/300,$ $~ 1500/450,$ $~ 1500/600, \dots,$ $~ 1500/150n, \dots$

                       2O m/s          1O m/s
                     3OOOm/15Os      15OOm/15Os       6 2/3 m/s
                         .              .            3OOOm/45Os
                        .             .            .
         |             .            .           .
   3OOOm |        red .       red .       red.          5 m/s
         |           .          .         .           15OOm/3OOs
         |          .         .        .          . '
         |         .        .       .         . '
         |        .       .      .        . '        3 1/3 m/s
         |       .      .     .       . '           15OOm/45Os
   15OOm |      . red .    .  red . '     red .  '
         |     .    .   .     . '       .  '
         |    .   .  .    . '     .  '
         |   .  . .   . '   .  '
         |  . .   . ' .  '
         | .. . '  '
         |__________________________________________
        O  green |red| green |red| green  red|
            1.5  | 1 |       |   |           |
            min  |min|       |   |           |
                9O   15O   24O   3OO         45O
               sec   sec   sec   sec         sec

The additional “speed” $~\nu=\infty$ m/s$~$ is a degenerate solution, passing through all lights while they are simultaneously green. (In the physical world, however, simultaneity itself is not uniquely defined.)

The speeds $~\nu = 20$ m/s, $~6 \frac{2}{3}$ m/s$~$ would be extra fun to drive because they alternately get the beginnings and ends of green lights. Even more speeds would be possible if greens lasted even longer relative to reds.

The limit where lights are virtually always green resembles Euclid's orchard, which relates to straight lines sneaking through a grid of points.