The farmer and the pig (Posted on 2008-11-13)

A farmer wants to catch a pig which is 100 meters away from him. They start running at the same time. The pig runs straight ahead, initially perpendicular to the line farmer-pig, at constant speed. The farmer always runs in the direction where he sees the pig (also at constant speed).

If the ratio of the speeds is 3:1 (farmer:pig), how far would the pig have traveled till the farmer caught it?

	Submitted by pcbouhid
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According to the drawing, let the pig start at O (the origin) and move along the x axis. The farmer starts at point B on the y axis such that OB = b = 100. At time t the pig will have moved to the point T on the x-axis, and the farmer will have moved along the arc shown, such that the tangent to the arc at the point the farmer is, is directed at the point T. Let s = arc length of BP. Because of the relative speeds, the arc BP will be three times the length of OT [arc BP = 3(OT)]. Draw the perpendicular PN from P to the x axis. If (x,y) are the coordinates of the farmer, we have NP/NT = -dy/dx since the slope of curve is given by NP/NT and the slope is negative. Then OT = ON + NT = ON + NP(NT/NP) = x - y(dx/dy) (note dx/dy not dy/dx) OT = (1/3)s = x - y(dx/dy) s = 3(x-y(dx/dy)) If we differentiate with respect to y: ds/dy = 3(dx/dy - dx/dy - y.d2x/dy2) = - 3y * d2x/dy2 But ds/dy = -sqrt(1+(dx/dy)2), the minus being taken because y decreases as s increases. If we write q for dx/dy this gives: sqrt(1+q2) = 3y * dq/dy dy/3y = dq/sqrt(1+q2) Integrating: (1/3)ln(ky) = sinh^(-1)(q) (k is constant of integration) When y = b, q = 0, so k*b = 1 or k = 1/b. Therefore, q = sinh[ln(y/b)(1/3)] sinh(x) = (ex - e-x)/2 2q = (y/b)(1/3) - (y/b)(-1/3) Since q = dx/dy a second integration gives: 2(x+c) = (3/4) * b(-1/3) * y(4/3) - (3/2) * b(1/3) * y(2/3) When x = 0, y = b, so that 2c = -(3/4)b and the equation of the curve becomes: b(1/3) * (8x-3b) = 3(y(4/3) -2b(2/3) * y(2/3)) The farmer catches the pig when y = 0, that is when 8x-3b = 0 or x = (3/8)b or x = 3/8 * 100 = 37,5m.

	Subject	Author	Date
	Puzzle Answer	K Sengupta	2022-08-24 00:31:32
	re(2): Analytical Solution	Daniel	2008-11-19 09:27:55
	re: Analytical Solution	Charlie	2008-11-17 16:06:52
	Analytical Solution	Daniel	2008-11-17 14:43:48
	Steps to solve- to lazy to complete	alex	2008-11-14 09:02:57
	numerical integration (spoiler)	Charlie	2008-11-13 18:43:08