Here is a diagram of the problem.
Where Should a Pilot Start Descent?
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 l from touchdown at the origin.
ii. The pilot must maintain a constant horizontal speed v
iii. The of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity).
1. Find a cubic P(x) = ax^3 + bx^2 + cx + d that satisfies condition (i) by imposing suitable conditions on P(x) and P'(x) at the start of descent and touchdown.
2. Use conditions (ii) and (iii) to show that .
i completed the first part, and i got p(x) = -2hx^3/l^3 + 3hx^2/l^2, but im a bit confused about the second part. I know how to get the 6h/l^2 but i dont know how to place the v^2 (or how it even got there).
to get the 6h/l^2, i simply took the third derivative of the polynomial p''(x) = 6ax + 2b, and i subbed in my a and b values from question one to get
p''(x) = 6(-2h/l^3)x + 2(3h/l^2)
i then subbed in x=0, since that is when the maximum accelreation will occur, and this value cannot be greater than a value k,
p''(0) = 6h/l^2
i dont know how i would include the v^2,