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Application of derivatives, to an aircraft landing.

An approach path for an aircraft landing is shown in the figure and satisfies the following conditions:

( i ) The cruising altitude is when descent starts at a horizontal distance from touchdown at the origin.

( ii ) The pilot must maintain a constant horizontal speed throughout descent.

( iii ) The absolute value of the vertical acceleration should not exceed a constant (which is much less than the acceleration due to gravity).

http://i.stack.imgur.com/ImsaT.png

- Find a cubic polynomial
*P*(*x*)=*a**x*3+*b**x*2+*c**x*+*d* that satisfies condition ( i) by imposing suitable conditions on P(x) and P'(x) at the start of descent and at touchdown.

WE know from the understanding of the graph that (0,0) is 1 point in the graph, so we can conclude that **d = 0**

*P*′(*x*)=3*a**x*2+2*b**x*+*c*

I am not sure if we can substitute (0,0) here as well.

Using P'(0) = 0, we can easily see that **c=0**.

So now we currently have *P*(*x*)=*a**x*3+*b**x*2

Now if we take P'(l) = 0 then we get l=0 or *l*=−*b*3*a*2

How do we use this to get rid of the **a** and **b** in the equation?

Re: Application of derivatives, to an aircraft landing.

Quote:

Originally Posted by

**Goku** An approach path for an aircraft landing is shown in the figure and satisfies the following conditions:

( i ) The cruising altitude is when descent starts at a horizontal distance from touchdown at the origin.

( ii ) The pilot must maintain a constant horizontal speed throughout descent.

( iii ) The absolute value of the vertical acceleration should not exceed a constant (which is much less than the acceleration due to gravity). http://i.stack.imgur.com/ImsaT.png

- Find a cubic polynomial
*P*(*x*)=*a**x*3+*b**x*2+*c**x*+*d* that satisfies condition ( i) by imposing suitable conditions on P(x) and P'(x) at the start of descent and at touchdown.

WE know from the understanding of the graph that (0,0) is 1 point in the graph, so we can conclude that **d = 0** *P*′(*x*)=3*a**x*2+2*b**x*+*c* I am not sure if we can substitute (0,0) here as well. Using P'(0) = 0, we can easily see that **c=0**. So now we currently have *P*(*x*)=*a**x*3+*b**x*2 Now if we take P'(l) = 0 then we get l=0 or *l*=−*b*3*a*2 How do we use this to get rid of the **a** and **b** in the equation?

You wrote

**Now if we take P'(l) = 0 then we get l=0 or l=−b3a2**

That's the wrong method: l is a known value and you are looking for values of a or b wrt l:

$\displaystyle P'(l)=3al^2+2bl=0~\implies~b=-\frac32 l \cdot a$

That means:

$\displaystyle P(x)=a x^3- \frac32 l \cdot a x^2 = ax^2 \left(x-\frac32 l \right)$