# Thread: Where should a pilot start descent

1. ## Where should a pilot start descent

Here is a diagram of the problem.

http://www.stewartcalculus.com/data/...p_0205_stu.pdf

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
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) = 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 .
(6hv^2)/l^2?k

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,

thanks,

2. ## Re: Where should a pilot start descent

Originally Posted by evox
Here is a diagram of the problem.

http://www.stewartcalculus.com/data/...p_0205_stu.pdf

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
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) = 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 .
(6hv^2)/l^2?k

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,

thanks,
This is a classic problem

http://www.freemathhelp.com/forum/viewt ... =3&t=26449

However, do not be tempted to copy the answers - your instructor might be looking at these reolies too.

3. ## Re: Where should a pilot start descent

Does my answer to question 1 look correct? it seems it is correct because if i sub in x=l, i get h and thats satisfies condition i

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