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dharmatech/pse-pro-4.17.txt Created Feb 7, 2012

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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