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// 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 |
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