June 2001
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