How far does the messenger travel?

Question

An army column begins to march. A messenger leaves the rear and delivers a message to the head of the column. With no turn around delay, he returns to the rear, arriving there when the rear just reaches where the head of the column started from.

Answer 1

Twice the length of the column

Because:
if C is the length of the column
Vc is the speed of Column
Vm is the speed of messenger

Da=Vc * t1
Vm*t1= C + Da
Da is the distence traveld by column in time t1 (messenger reaches head)

similarly
Db=Vc *t2
Vm*t2=C-Db

Db is the distence traveld by messenger to reach the rear (== previous head)

Da=Db (no turn arround time)

Distence traveld by messenger =Vm*t1 + Vm*t2
=C+Da+C-Db
=2C+Da -Da (Da=Db)
=2C

Answer 2

I've seen this question before, but you need the size of the column for an exact answer. It has to do with speed, and time.

Answer 3

length of the army is La
rate of messenger is Rm
rate of army movement is Ra
Messenger distance traveled is Dm
Army traveled distance is Da
t is time.

Total distance for army
Ra*t=Da
t=Da/Ra
But wait Da can also be said to be equal to
Da=La
So
t=La/Ra
Total distance for messenger
Rm*t=Dm
Time is the same in each equation so:
Rm(La/Ra)=Dm
We're not quite done but if you can establish the rate of the messenger to the length of the army relationship you should come up with your answer.

Answer 4

Are they marching in a straight line? A circular formation or some other shape depending upon the open space available for maneuvers? I can't answer this one very well.

;-)