Where Should a Pilot Start Descent?

In: Science

Submitted By we7524
Words 838
Pages 4
* A. * * 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 touch- down at the origin. (Note: in this coordinate system, the direction of flight is from right to left or in the negative –x direction).

(ii) The pilot must maintain a constant horizontal speed v throughout the descent.(Note: speed is the absolute value of velocity. In this instance the velocity will be negative, however, the speed will be a positive number of the same magnitude).

(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 that satisfies condition (i) by imposing suitable conditions on and at the start of descent and touchdown. *
Solution: Assume the cubic polynomial function is P(x)= ax3 + bx2 + cx + d
According to the graph, We know P(0) = 0, P'(0) = 0, P(L) = h, P'(L) = 0
The original equation is P(x)= ax3 + bx2 + cx + d
P(0) = 0, so plug in zero for x. You will find that d = 0.
Simplify the equation to P(x)= ax3 + bx2 + cx
Find the derivative of P(X), which is 3ax2 + 2bx + c
P’(0) = 0, so plug in zero for x. We found that c =0
Simplify the equation to P’(X) = 3ax2 + 2bx
Plug in P'(L) = 0
We get the equation of 3aL2+ 2bL= 0 ①
Plug in P(L) = h
We get the equation of aL3 + bL2 = h ②
Solve for a,b
3aL2+ 2bL= 0 ①aL3 + bL2 = h ②

As a result, P(X) = (-2hL3) x3 + (3h2L2) x2

* * 2. Use conditions (ii) and (iii) to show that: 6hv2L2 ≦ K

Steps 1. Find the first implicit derivative of P(L) dPdt= 3aL2( dxdt)+2b(dxdt) dxdt = -v 2. Find the second implicit derivative…...

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