Ponder This Challenge - June 2001 - Goblin chase in a train

Ponder This Challenge:

(Based on a puzzle of the nineteenth century by George Lilley, brought to our attention by Amit Elazari.)

This month we once again find a goblin chasing us.

This time we are on a train, heading due east at 60 kilometers per hour.

The goblin starts out 30 kilometers due north of us.
He runs at a constant speed, but (fortunately for us) has lost the ability to plan ahead; he always heads directly towards our current position.

(A clever goblin would aim in front of us, since our train's future motion is so predictable; but our goblin is not clever.) He catches up to us after we have travelled 100 kilometers (that is, after 100 minutes).

How fast does the goblin move?

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Let t denote time in hours.
  Let (x,y) be the goblin's current position, with our initial position being (0,0).
  Let u=60 be our speed, v the goblin's unknown speed, and r=u/v their ratio.
  Let the goblin's initial position be (0,g) where g=30.
  Let "^" denote exponentiation (as in LaTeX).

  The goblin's motion satisfies the differential equations:

  (dx/dt)^2 + (dy/dt)^2 = v^2 (the goblin moves at speed v);

  x - y dx/dy = ut (he is always pointing at our current position).

  The initial conditions: at y=g, x=t=0.

  The solution: the goblin's position (x,y) at time t is given parametrically in terms of y:

  t = (g/(v(1-r^2)) - (g/2v) ( (y/g)^(1+r)/(1+r) + (y/g)^(1-r)/(1-r) ).

  x = (gr/(1-r^2)) + (g/2) ( (y/g)^(1+r)/(1+r) - (y/g)^(1-r)/(1-r) ).

  One can calculate dt/dy and dx/dy, and verify that the differential equations are satisfied.

  When y=g we get t=0 and x=0, which meets the initial conditions.
  When y=0 we get t=g/(v(1-r^2)) and x=gr/(1-r^2).

  The last equation gives 100=30r/(1-r^2), whence r=(sqrt(409)-3)/20 and v=60/r=3*(sqrt(409)+3)=69.67124525 km/hour.

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Marian Olteanu (06.01.2001 @ 11:48 AM EDT)
• Henry Bottomley (06.01.2001 @ 1:58 PM EDT)
• Hareendra Yalamanchili (06.01.2001 @ 1:28 PM EDT)
• Vladimir Sedach (06.01.2001 @ 1:56 PM EDT)
• Shmuel Spiegel (06.01.2001 @ 2:39 PM EDT)
• Bill Schwennicke (06.01.2001 @ 3:10 PM EDT)
• Ether Jones (06.01.2001 @ 9:55 PM EDT)
• Yuriy Groysman (06.02.2001 @ 5:56 AM EDT)
• Dharmashankar Subramanian (06.02.2001 @ 8:17 PM EDT)
• Saibal Mitra (06.02.2001 @ 9:50 AM EDT)
• Vince Lynch (06.02.2001 @ 11:07 AM EDT)
• Sudipta Das (06.02.2001 @ 12:12 PM EDT)
• Jacques Willekens (06.03.2001 @ 7:27 AM EDT)
• Bill Webb (06.03.2001 @ 10:39 PM EDT)
• Adam Daire (06.04.2001 @ 8:00 PM EDT)
• Pritibhushan Sinha (06.05.2001 @ 9:31 PM EDT)
• Robert Sides (06.05.2001 @ 10:03 PM EDT)
• Vinko Marinkovic (06.05.2001 @ 12:34 PM EDT)
• Bill Wepp (06.05.2001 @ 5:19 PM EDT)
• Patrice De Pra (06.06.2001 @ 3:59 AM EDT)
• Krishna Kumar (06.07.2001 @ 7:26 AM EDT)
• Liz Savery (06.11.2001 @ 6:31 PM EDT)
• liuwei (06.11.2001 @ 6:54 PM EDT)
• Artus Hervé (06.12.2001 @ 10:32 AM EDT)
• Jim Callen (06.12.2001 @ 2:42 PM EDT)
• Stephane Higueret (06.14.2001 @ 8:34 AM EDT)
• Steve Clymer (06.14.2001 @ 1:20 PM EDT)
• Mark Pilloff (06.17.2001 @ 9:26 PM EDT)
• Alec Klatchko (06.18.2001 @ 2:00 AM EDT)
• Michel Jacquemin (06.20.2001 @ 4:57 AM EDT)
• David Friedman (06.20.2001 @ 10:54 PM EDT)
• Douglas Schrag (06.23.2001 @ 8:45 PM EDT)
• Alexey Vorobyov (06.22.2001 @ 4:04 AM EDT)
• Larry Kyrala (06.25.2001 @ 3:27 AM EDT)
• gsd+cas+gdn (06.25.2001 @ 8:15 AM EDT)
• Dave Biggar (06.26.2001 @ 10:10 AM EDT)
• Jan van Delden (06.27.2001 @ 9:51 AM EDT)
• Santhosh Bandreddi (06.27.2001 @ 5:33 PM EDT)
• Suresh Kodati (06.28.2001 @ 8:19 AM EDT)
• Ben Rapp (06.28.2001 @ 12:43 PM EDT)
• Prithu (06.29.2001 @ 1:18 PM EDT)