Angry Lion

You see an angry, smart lion to the east. It's angry because you just shot it in the leg with your last bullet, but it's still twice as fast as you are. The only chance for survival is to jump into the river that runs east-west some distance to the north from you. Unfortunately the lion is smart and won't follow a pursuit curve of heading for you directly, but towards the point in which it will catch you if you keep running in a straight line. In which direction should you run to get as far north as possible before the lion catches you, thus maximizing your chances of survival by reaching the river before the lion catches you?

straight to the river.

Originally posted by talzamir
You see an angry, smart lion to the east. It's angry because you just shot it in the leg with your last bullet, but it's still twice as fast as you are. The only chance for survival is to jump into the river that runs east-west some distance to the north from you. Unfortunately the lion is smart and won't follow a pursuit curve of heading for you directly, ...[text shortened]... , thus maximizing your chances of survival by reaching the river before the lion catches you?

South

After a polar circumnavigation you end up on the opposite side of the river to the wounded lion!

A lovely answer. ^_^

Here's mine. A bit boring compared to heading south, but still.

Both you and the lion run in a straight line - you because that's how you get farthest from the your starting point, and the lion because it heads for the interception point. If the interception point is in the river or beyond it, you win. If not, you join the lion for lunch.

The speeds are irrelevant in this, as long as the ratio is fixed, here 2:1. The distance doesn't change the optimal direction either, only how far you get along that course. To make this more legible, let's say the lion starts at point (300, 0) and you at point (0,0) and the interception point is a point (x,y) where y > 0 (you head more north than south, treating the planet as big enough when compared with rifle range) that is twice as far from the lion's starting point as it is from yours. You get, using Pythagoras,

2 (x^2 + y^2)^(1/2) = ((x-300)^2+y^2)^(1/2)

Which can be expressed as

(x + 100)^2 + y^2 = 200^2

...which is the equation of a circle with the center point at (-100,0) and a radius of 200. Running in a straight line you can reach any point inside the circle faster than the lion, and point on the circumference at the same time as the lion, and barring miracle or river, there is no way to get out of the circle alive. Your best bet is to head in the direction of the point (-100,200).. like a chess knight. That way you get 200 yards north; heading straight north right from the start you get 27 yards less. The bigger the difference in velocity between you and the lion is, the better north looks as direction - the smaller the difference, the better west looks.