Where should a Pilot Start Descent?

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An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: i. The cruising altitude is h when descent starts at a horizontal distance I from touchdown at the origin. ii. The pilot must maintain a constant horizontal speed v throughout descent. iii. The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity.) a. Find a cubic polynomial P(x) = ax3 + bx2 + cx + d (where x is the horizontal distance from the airport and P(x) is the corresponding altitude of the plane) by considering the following conditions on P(x) and P'(x). (Note: PM is not the velocity because P is not a function of time. P'(x) is change in altitude divided by change in distance from the airport.) P(l) = h, P(O) = 0, P'(l) = 0, P'(0) = 0 Note: Since h and l are constants, our coefficients a, b, c, and d will be in terms of h and l. 6hv2 12 _ b. Use conditions ii and iii to show that Note: This part involves related rates. dx 71-; (the horizontal speed) is given in condition ii. We have P(x) = y so we have V as a function of x. d2 Differentiate implicitly to find ET: (the vertical acceleration) Hint - the maximum value for the absolute value of the vertical acceleration will occur when x = l. c. Suppose that an airline decides not to allow vertical acceleration of a plane to exceed k = 860 nri' if the cruising altitude of a plane is 35,000 ft and the horizontal speed is 300 mi/hr, how far away from the airport should the pilot start the descent? d. Graph the approach path if the conditions stated in Problem 3 are satisfied