10 - Messenger Going From Front To Rear Then To Front Of A Marching Army Column

Problem
An army of troops is marching along a road at 5 kph. A messenger on horseback was sent from the front to the rear of the column and returns immediately back. The total time taken being 10 minutes. Assuming the messenger rides at the rate of 10 kph, determine the length of the column.

Answer Key

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[ 625 m ]

Solution

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Let x = length of the army column in km

Going from front to rear:

relative velocity = 10 + 5 = 15 kph
relative distance traveled = x km

Going from rear to front:

relative velocity = 10 - 5 = 5 kph
relative distance traveled = x km

Note: time = distance / velocity
$t_\text{front to rear} + t_\text{rear to front} = 10 ~ \text{minutes}$

$\dfrac{x}{15} + \dfrac{x}{5} = \dfrac{10}{60}$

$\dfrac{4}{15}x = \dfrac{1}{6}$

$x = 0.625 ~ \text{km}$

$x = 625 ~ \text{m}$ ← answer

10 - Messenger Going From Front To Rear Then To Front Of A Marching Army Column

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