How often do the buses run?

Question

A cyclist rides at a constant speed along a straight road served by buses. Buses run at a constant speed in both directions, leaving at regular intervals that are the same for each direction. The cyclist notices that a bus overtakes him from behind every 12 minutes, and that he meets an oncoming bus every 4 minutes.

How often do the buses leave, and how does the bus speed compare with the cyclist's?

Accepted Answer

Buses leave every 6 minutes, and they travel at twice the cyclist's speed. Set up. Let the buses travel at speed v and the cyclist at speed c. Buses in one direction leave every T minutes, so consecutive buses are a distance vT apart. - Overtaking. A bus behind the cyclist closes the gap at speed v minus c. Successive overtakings are one spacing apart: vT / (v minus c) = 12. - Oncoming. An approaching bus closes at v + c: vT / (v + c) = 4. Solve. From the two equations, 12(v minus c) = 4(v + c), so 12v minus 12c = 4v + 4c, so 8v = 16c and v = 2c: the buses go twice as fast as the cyclist. Sub…

How often do the buses run?

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