Hathaway's pursuit curve
This activity belongs to the GeoGebra book The Domain of the Time.
This problem was posed by the American mathematician A. S. Hathaway in 1920.
A dog in the center of a circular pond chases a duck swimming along the edge of the pond. If the dog swims n times faster than the duck, determine the equation of the pursuit curve and the distance the dog swims until it catches the duck.
Hathaway could not find the equation of the curve. Indeed, the resulting differential equation does not have an analytical solution (it can only be solved numerically).
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Move M (duck) and N (dog) and stop the animation when N and M are close enough
SetValue(M, M + dt vM)
SetValue(N, N + dt vN)
StartAnimation(anima, abs(N − M) > (x(Corner(2) − Corner(1))/400))
# Use the polyline as a trace by recording the position N in the list reg
SetValue(reg, Append(N, reg))
Author of the activity and GeoGebra construction: Rafael Losada.