IBM Research | Ponder This | June 2001 solutions

# Ponder This

## June 2001

<<May June July>>

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.

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