A body of soldiers form a 50m-by-50m square ABCD on the parade ground.

In a unit of time, they march forward 50m in formation to take up the position DCEF.

The army's mascot, a small dog, is standing next to its handler at location A. When the soldiers start marching, the dog begins to run around the moving body in a clockwise direction, keeping as close to it as possible. When one unit of time has elapsed, the dog has made one complete circuit and has got back to its handler, who is now at location D (assume the dog runs at a constant speed and does not delay when turning the corners).

 B----C----E
 |    |    |   
 A----D----F

How far does the dog travel?

L=211.8 m

u = speed of the soldier
v= speed of the dog
a=u/v (a<1)
b= the distance to the new position (50m)

The 1-st and the 3-rd path is the length of the pursuit curve:
L1=L3=b/(1-a²)

L2=b/(1-a)
L4=b/(1+a)

the equation of time for both dog and soldiers gives the value of "a":
2/(1-a²)+1/(1-a)+1/(1+a)=1/a
a²+4a-1=0
a=√5-2

L=L1+L2+L3+L4 = 2b/(1-a²)+b/(1-a)+b/(1+a)
L=50(2+√5)=211.8m

Posted by luminita on 2004-01-05 05:20:17