Maximum path length of a projectile (Adapted from Putnam Exam 1940) A projectile is launched from the ground with an initial speed V at an angle u from the horizontal. Assume that the x-axis is the horizontal ground and y is the height above the ground. Neglecting air resistance and letting g be the acceleration due to gravity, it can be shown that the trajectory of the projectile is given by y = - 1 2 kx2 + ymax, where k = g 1V cos u22 and ymax = 1V sin u22 2g . a. Note that the high point of the trajectory occurs at 10, ymax2. If the projectile is on the ground at 1-a, 02 and 1a, 02, what is a? b. Show that the length of the trajectory (arc length) is 21 a 0 21 + k2x2 dx. c. Evaluate the arc length integral and express your result in terms of V, g, and u. d. For a fixed value of V and g, show that the launch angle u that maximizes the length of the trajectory satisfies 1sin u2 ln 1sec u + tan u2 = 1. e. Use a graphing utility to approximate the optimal launch angle.

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