The Ultimate Bus

Buses are a nation's pride in some countries and a disgrace in others. Math questions can handle random behavior like the bus schedule in some countries, but it's much better to deal with a good bus system, with a normal schedule. Here is one question about such system.

Andy walks at a constant speed along a long street. Every 6 minutes he is passed by a bus going in this direction. Every 2 minutes he is also passed by a bus going in the opposite direction. Both directions have the same schedule and all buses go at the same speed. What is the time difference between two consecutive bus departures?

First we need to define some variables and understand a conecpt of of physics. Let's call the speeds of Andy and the buses a and b, respectively.

A relative speed is the speed at which two distinct objects move towards or away from each other. This speed is given by subtracting the two speeds. In the first case, the bus is approaching Andy at a speed of b - a. Since the other buses go in the opposite direction, their speed becomes "negative", and the bus goes towards Andy at a speed of b + a.

Let x be the distance between the buses (they are scheduled on regular intervals, so the distance between buses is the same for all buses)

Using the formula distance = velocity * time, we can say that:
x = (b - a) * 6
x = (b + a) * 2

Equate the x's:
6b - 6a = 2b + 2a
4b = 8a
b = 2a
a = b/2

Either Andy is very fast of the buses are very slow, because the buses go at a speed just twice as Andy. That is definitely weird.

x = (b - b/2) * 6 = b / 2 * 6 = 3b

Since the buses need to travel a distance of 3b to get from the location of one bus to another bus and they do so at a speed of b, a bus must be sent every 3b / b = 3 minutes. That's a great interval for the passengers of this bus system.

Hope you liked it,
Nadav

nadavs