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.

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