Chapter 3.4, Problem 1AP

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 $\mathcal{l}$ 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).

1. Find a cubic polynomial $P(x)=a{x}^3+b{x}^2+cx+d$ that satisfies condition (i) by imposing suitable conditions on $P(x)$ and $P^{\prime }(x)$ at the start of descent and at touchdown.