Pilot Descent Point, Derivatives

[brembo]

Homework Statement

Where should the pilot start descent? The approach path for an aircraft should satisfy:
i) The cruising altitude is h when descent starts. At horizontal distance.
ii) The pilot must keep a constant horizontal velocity, Vx, throughout the decent.
iii)|αy|≤ K << g
k = constant
g = gravity
|αy| = vertical component of acceleration

1) Find a cubic polynomial
y = P(x) = ax³ + bx² + cx + d that satisfies:
(i) on P(x) , P'(x) @ x = 0 , x = L

2) Use (i) , (ii) to show:
(6hV_x²) / L² ≤ k

Homework Equations

P(x) = ax^3 + bx^2 + cx + d
(6hVx^2) / L^2 ≤ k

The Attempt at a Solution

I was able to conclude that:

y(0) = P(0) = d = 0 → P(x) = ax³ + bx² + cx
And @ x = L : P(L) = aL³ + bL² + cL = y(L) = h



Thats all I was able to get so far. I am stuck. If you guys need anymore info on the question let me know and ill do my best to help out

Thanks!

 

[mighty2000]

Thank you for your question. It seems that you are on the right track in finding a cubic polynomial that satisfies the given conditions. However, to solve for the coefficients a, b, and c, we need to use the information provided in (ii) and (iii).

First, let's rewrite P(x) as a function of time, t, instead of distance, x. We can do this by using the constant horizontal velocity, Vx, as follows:

P(t) = at^3 + bt^2 + ct

Next, let's consider the vertical component of acceleration, αy, which is given by:

αy = d^2y/dt^2 = 2at + b

Since we want |αy|≤K, we can set up the following inequality:

- K ≤ αy ≤ K

- K ≤ 2at + b ≤ K

- K - 2at ≤ b ≤ K + 2at

Now, we can use this inequality to solve for the coefficients a, b, and c. Substituting t = 0 and t = L into P(t) and using the given information, we can set up the following system of equations:

P(0) = 0 = c

P(L) = h = aL^3 + bL^2

K - 2a(0)t ≤ b ≤ K + 2a(0)t

K - 2aLt ≤ b ≤ K + 2aLt

Using the first two equations, we can solve for a and b in terms of L and h:

a = (h - bL^2)/L^3

b = h - aL^3

Substituting these values into the third equation, we get:

K - 2(h - aL^3)Lt/L^3 ≤ b ≤ K + 2(h - aL^3)Lt/L^3

K - 2(h - (h - bL^2)L^3/L^3)Lt/L^3 ≤ b ≤ K + 2(h - (h - bL^2)L^3/L^3)Lt/L^3

K - 2hLt/L^2 ≤ b ≤ K + 2hLt/L^2

Now, we can substitute these values for a, b, and c back into our original polynomial equation P