There are parallel train tracks running from Abilene to Imogene. The tracks run in a straight line between the two cities. On a particular morning, Train A begins the journey from Abilene to Imogene on the first track. At the exact same time, Train B begins the journey from Imogene to Abilene on the second track. The two trains always travel at constant velocities.

The two trains pass each other at a small town called Xanadu. From there, Train A takes another 5 hours to reach Imogene, while Train B takes another 10 hours to reach Abilene.

The next day, Train A embarks on the return trip to Abilene at the exact same time Train B leaves for Imogene. This time, they pass each other at a small town called Yonkers, whereby Train A takes another 5 hours to reach Abilene while Train B takes another 10 hours to reach Imogene.

If the distance between Yonkers and Xanadu is 500 miles, then how fast is each train traveling?

    A : Abilene
    I : Imogene  
    X : Xanadu
    Y : Yonkers
    t : Time to reach X = Time to reach Y
 v(A) : Velocity of train A
 v(B) : Velocity of train B

   |AX| = t*v(A) = 10*v(B)

   |XI| = t*v(B) = 5*v(A)

Solving these gives

   v(A) = v(B)*sqrt(2)

   t = 5*sqrt(2)

Since the description of the trips is the same,
we must conclude that |AY| = |XI|.

   |AX| = |AY| + |YX| = |XI| + |YX|

               or

   t*v(A) = 5*v(A) + 500

               or

   5*sqrt(2)*v(A) = 5*v(A) + 500

               or

   v(A) = 100*(1+sqrt(2)) ~= 241.42 miles/hour

               and

   v(B) = 50*(2+sqrt(2)) ~= 170.71 miles/hour   

Posted by Bractals on 2007-05-05 11:26:22