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Outfoxing a fox (Posted on 2019-02-23) Difficulty: 3 of 5
A rabbit is trying to avoid detection by a nearby fox.

Initially, the fox and the rabbit are out of each other's sights as they are on opposite sides of a tree at the origin with the rabbit 1 unit North and the fox 1 unit South.

The fox starts to run directly to the left in a straight line with a constant speed 1 unit.

To avoid detection, the rabbit starts to run along a certain path with a constant speed 2 units, so as to always keep the tree between itself and the fox.

Find the differential equation which describes the path (in polar coordinates) of the hiding rabbit.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution | Comment 1 of 2

Let a be the polar angle, R the distance of the rabbit from the tree at time T, and let b be the angle between the rabbit’s original position and the tree, measured from the rabbit’s position at time T. (I’ve drawn a figure, but don’t know how to enter it!)

After time T, the fox has travelled to the left a distance T, so T = cot a. The rabbit has travelled a distance 2cot a; with the fox, tree and rabbit lying on a line. 

The 3 points: rabbit's position(time=0), rabbit's position(time=T) and origin make up a triangle, in which (b, 1) and (pi/2 - a, 2cot a) are pairs of angles and opposite sides. The law of sines then gives: sin a = 2 sin b.

From the law of cosines we get: R^2 = 1 + 4cot^2 a + 4cot a sin(a-b). So:

db/da = cos a/(2 cos b) = cos a / sqrt( 4 - sin^2 a)

R dR/da = -2 cot a csc^2 a - 2 csc^2 a sin(a-b) + 2 cot a cos(a-b)( 1 - cos a / sqrt( 4 - sin^2 a) )

Which (if I haven’t made any mistakes) gives an unwieldy differential equation. Presumably it could be simplified, but this solves the problem.

Edited on February 24, 2019, 8:31 pm
  Posted by FrankM on 2019-02-24 19:56:38

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