An aircraft begins its descent at a distance x = L (x = 0 is the spot at which t

Question

An aircraft begins its descent at a distance x = L (x = 0 is the spot at which the plane touches down) and an altitude of H. Suppose a cubic polynomial of the following form is used to describe the landing: y = ax^3 + bx^2 + cx + d where y is the altitude and x is the horizontal distance to the aircraft. The aircraft begins its descent from a level position, and lands at a level position. Solve for the coefficients a, b, c, and d. If the aircraft maintains a constant forward speed (dx/dt = u = constant) and the magnitude of the vertical acceleration ( -%) is not to exceed a constant A, show that . If A = 0.3 ft/s^2, H = 15000 ft, and u = 200 mph, how far from the airport should the pilot begin the descent?

Explanation / Answer

Y(x)=ax^3+bx^2+cx+f

Y(0)=0 so d=0

Y(1)=h=al^3+bl^2+cl=h

Constant horizontal speed so,dx/dt=-y

dy/dt=dy(x)/dx dt dx/dt= -u(3ax^2+2bx+c)

When x=0 dy/dt=0 so c=0

Assume when x=l,dy/dt=0

So,3al^2+2bl=0(2)

al^3+bl^2=h(3) Solve 2 and 3

a=-2h/l^3,b=3h/l^2

b)vertical accelaration is d^2y/dt^2=d(dy/dt)/dt

=d(-u(3ax^2+2bx))/dx *dx/dt=u^2(6ax+b) Lessthan A

6h/l^2-12hx/l^3 lessthan A/u^2

When x=0 6hu^2/l^2 lessthan A

C) A=0.3ft/s^2,h=15000ft,u=200mph

Substitute values, l=293×squareroot 6×15000/0.3=1627ft

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