Henry Ernest Dudeney/Puzzles and Curious Problems/210 - Pat and his Pig/Solution

by : $210$

 * Pat and his Pig

Solution
Pat catches the pig after it has run $66 \tfrac 2 3$ yards towards the gate.

Proof
This can be modelled using the technique of Differential Equation of Perpendicular Pursuit Curve.

Let us rotate the frame of reference so as to make:
 * the initial position of the pig at the origin of a Cartesian plane
 * the initial position of the farmer at the point $\tuple {c, 0}$ on this frame
 * the initial position of the gate at the point $\tuple {0, c}$ on this frame.

Let $P$ be the position of the pig at time $t$.

Let $F$ be the position of the farmer Pat at time $t$.

From Differential Equation of Perpendicular Pursuit Curve, the differential equation describing the path taken by $F$ is:


 * $\dfrac {\d y} {\d x} = \dfrac 1 2 \paren {\paren {\dfrac x c}^{1 / 2} - \paren {\dfrac c x}^{1 / 2} }$

So we need to solve this for a start.

Simplification is in order.

Thus we have:

This differential equation can be solved by Separation of Variables:


 * $\ds \int \rd y = \dfrac 1 {2 \sqrt c} \int \dfrac {x - c} {\sqrt x} \rd x + C$

When $y = 0$ we have that $x = c$ and so:

leaving us with:


 * $y = \dfrac {1 \sqrt x^3} {3 \sqrt c} + \sqrt {c x} + \dfrac {2 c} 3$

The farmer catches his pig when $x = 0$, that is:


 * $y = 0 + 0 + \dfrac {2 c} 3$

That is, $\dfrac 2 3$ of the way to the gate, or $66 \tfrac 2 3$ yards from $B$.