(Miracle on the Hudson) Airline pilot Sully Sullenberger is flying his Airbus A320 when a flock of geese fly into one of his engines. He needs to make an emergency landing and sees a point on the Hudson River, l meters away. At time t=0, his altitude is h meters and his flight trajectory is horizontal. In order to successfully land, he must.
1) Maintain a horizontal speed v throughout the flight.
2) Keep his vertical acceleration less than k at the time of landing so that his passengers don't feel sick.
3) Keep his flight trajectory exactly horizontal initially and when he lands.
a) Assuming that his flight trajectory is a cubic polynomial of the form y=ax^3+bx^2+cx+d, deduce the relations a = -2h/l^3, b = 3h/l^2, c = d = 0.
b) If the position of his plane is (x(t),y(t), show that x(t) = l-vt, y''(t) is less than k.
c) By differentiating the flight trajectory y = ax^3+bx^2+cx+d with respect to t (not x) and using your calculations from above, deduce that Captain Sullenberger must fly with horizontal speed v satisfying 6hv^2/l^2 is less than k.
Thanks for the help!