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Sunday, July 27, 2008

Spirited Soldier

A military column of 1 km length marches with a constant speed of 6 kilometers per hour. A courier from the back of the column is sent to the head of the column by bike to deliver a message. Arrived at the front, he instantly returns. When he returns again at the back, the column has travelled a distance of 1 km since the start of his errand. The courier drove with constant speed.

The Question: How fast did the courier bike?

Answer: The relative speed of the courier (compared to the column) is v-6 on the journey to the front, and v+6 on the way back. The distance travelled (relatively to the column) by the courier is in both cases 1 km, therefore the total time he is biking is

1/v-6 + 1/v+6

The column travels 1 km in 10 minutes (1/6 hour), therefore

1/v-6 + 1/v+6 = 1/6

This can be written as a quadratic equation (v+6) + (v-6) = (v-6) × (v+6) / 6 with as only allowed solution v = 6 + 6 × sqrt(2) km/h (approximately 14,49 km/h).