/*
 * An object begins a fall from position (1000,0), sliding along a frictionless
 * guide to position (px,py). It then slides along another guide
 * from (px,py) to position (0,1000).  Find px and py that minimize descent time.
 * The minimum value of the function found by Nonlin is the descent time in
 * seconds.  All coordinates are in centimeters.
 */
	Title "Two segment path for fastest descent";
	Parameter px;		// X coordinate of bend
	Parameter py;		// Y coordinate of bend
	Constrain px=.1,999;	// px must be in range 0 < px < d
	Constrain py=.1,999;	// py must be in range 0 < py < h
	Double G=980;		// Acceleration of gravity = 980 cm/sec^2
	Double sx=0, sy=1000;	// Starting x and y coordinate
	Double ex=1000, ey=0;	// Ending x and y coordinate
	Double d1,d2;		// Length of each segment
	Double a1,a2;		// Acceleration along each segment
	Double t1,t2;		// Fall time along each segment
	Double s1;		// Speed at end of segment 1
/*
 *  Determine length of each segment.
 */
	d1 = sqrt((px-sx)*(px-sx) + (py-sy)*(py-sy));
	d2 = sqrt((px-ex)*(px-ex) + (py-ey)*(py-ey));
/*
 *  Determine acceleration for each segment (proportional to slope).
 */
	a1 = G*(sy-py)/d1;
	a2 = G*(py-ey)/d2;
/*
 *  Determine time for segment 1 (starting speed is 0).
 */
	t1 = sqrt(2.*d1/a1);
/*
 *  Determine speed at end of segment 1.
 */
	s1 = a1 * t1;
/*
 *  Determine time for segment 2 (speed is s1 at start of segment).
 */
	t2 = (sqrt(s1*s1 + 2.*a2*d2) - s1) / a2;
/*
 *  Minimize the total fall time.
 */
	function t1 + t2;
	Data;