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).
- Find a cubic polynomialP(x)=ax3+bx2+cx+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)=3ax2+2bx+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)=ax3+bx2
Now if we take P'(l) = 0 then we get l=0 or l=−b3a2
How do we use this to get rid of the a and b in the equation?