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# The dog chases the fox

## Question

A dog hiding in the tall grass at the edge of a circular field spots a fox at the center of the eld. The fox, unable to see the dog but sensing the danger, begins to run at a constant speed along a straight line path towards the safety of a uniformly selected random point at the edge of the eld. At that same instant the dog jumps
the fence and runs straight at the fox at a constant speed M times that of the fox.
Find the probability that the fox will escape the field, and the value of M at which the dog must run to have a 50% chance of catching the fox.

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Got it. Similar to

Plot of dog's pursuit path.

Dog's speed ratio necessary to catch the fox, vs Fox's escape route angle.

Angle measured from x-axis, away from dog.

If fox chooses angle at random, half the time it will escape if

the dog's speed multiplier is the golden ratio 1.618...

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√2 ??

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Nice problem but just to clarify

Can the dog choose his path? I.e. could it be a straight line?

Must the dog maintain a heading directly toward the fox? I.e. must it be a curved path?

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The problem has a solution.

If M is very large, the dog catches the fox before he takes his first step.

If M > 2, the fox's best path away from the dog still gets him caught.

If M is vanishingly small, the fox's exit point must only lie outside a vanishingly small arc near the dog.

That is, for M in [0, 2], the escape probability is in [0, 1].

Wouldn't it be interesting if for M=1 p=0.5.

Off to solve an equation.

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