pse-pro-4.17.txt

        // A cannon with a muzzle speed of 1000 m/s is used to
        // start an avalanche on a mountain slope. The target is 
        // 2000 m from the cannon horizontally and 800 m above
        // the cannon. At what angle, above the horizontal, should
        // the cannon be fired?

        // PSE PRO 4.17

        // xi =    0
        // yi =    0
        // xf = 2000
        // yf =  800

        // vi = 1000

        // ax =  0
        // ay = -9.8

        
        // (2.3.1): xf = xi + vi cos(th) t + 1/2 ax t^2
        //          xf = vi cos(th) t
        // t        t = xf / (vi cos(th))       (2.3.1.1)

        // (2.4.1):     yf = yi + vi sin(th) t + 1/2 ay t^2
        //              yf = vi sin(th) t + 1/2 ay t^2
        // /. (2.3.1.1) yf = vi sin(th) {xf / (vi cos(th))} + 1/2 ay {xf / (vi cos(th))}^2
        //              yf = sin(th) xf / cos(th) + 1/2 ay xf^2 / (vi^2 cos^2(th))
        // * 2 cos^2(th)    2 cos^2(th) yf = 2 cos^2(th) sin(th) xf / cos(th) + 2 cos^2(th) 1/2 ay xf^2 / (vi^2 cos^2(th))
        //                  2 cos^2(th) yf = 2 cos(th) sin(th) xf + ay xf^2 / vi^2
        // power-reducing / half angle identity     cos^2(x) = [1 + cos(2x)]/2 :
        //                  2 [1 + cos(2 th)]/2 yf = 2 cos(th) sin(th) xf + ay xf^2 / vi^2
        //                  [1 + cos(2 th)] yf = 2 cos(th) sin(th) xf + ay xf^2 / vi^2
        // double angle formula 2 sin(x) cos(x) = sin(2x) :
        //                  [1 + cos(2 th)] yf = sin(2 th) xf + ay xf^2 / vi^2
        //                  yf + yf cos(2 th) = sin(2 th) xf + ay xf^2 / vi^2
        //                  sin(2 th) xf - yf cos(2 th) = yf - ay xf^2 / vi^2
        // xf = r cos(phi)
        // yf = r sin(phi)
        //                  sin(2 th) r cos(phi) - r sin(phi) cos(2 th) = yf - ay xf^2 / vi^2
        //                  r [ sin(2 th) cos(phi) - sin(phi) cos(2 th) ] = yf - ay xf^2 / vi^2
        // sum/difference identity  sin(x) cos(y) - sin(y) cos(x) = sin(x - y) :
        //                  r sin(2 th - phi) = yf - ay xf^2 / vi^2
        // r = sqrt(xf^2 + yf^2)
        //                  sqrt(xf^2 + yf^2) sin(2 th - phi) = yf - ay xf^2 / vi^2
        // tan(phi) = yf / xf       phi = arctan(yf / xf):
        //                  sqrt(xf^2 + yf^2) sin(2 th - arctan(yf / xf)) = yf - ay xf^2 / vi^2
        // sin(2 th - arctan(yf / xf)) = [yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)
        //
        // arcsin 1: 2 th - arctan(yf / xf) = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}         (arcsin1)
        //
        // and
        //
        // arcsin 2: 2 th - arctan(yf / xf) = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}    (arcsin2)

        // (arcsin1):
        //
        // 2 th - arctan(yf / xf) = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}
        // 2 th = arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)
        // th = {arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)} / 2

        // (arcsin2):
        //
        // 2 th - arctan(yf / xf) = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)}
        // 2 th = PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)
        // th = [PI - arcsin{[yf - ay xf^2 / vi^2] / sqrt(xf^2 + yf^2)} + arctan(yf / xf)] / 2