# Thread: Tough Calculus Problem...Can't figure out.

1. ## Tough Calculus Problem...Can't figure out.

(Miracle on the Hudson) Airline pilot Sully Sullenberger is flying his Airbus A320 when a flock of geese fly into one of his engines. He needs to make an emergency landing and sees a point on the Hudson River, l meters away. At time t=0, his altitude is h meters and his flight trajectory is horizontal. In order to successfully land, he must.

1) Maintain a horizontal speed v throughout the flight.
2) Keep his vertical acceleration less than k at the time of landing so that his passengers don't feel sick.
3) Keep his flight trajectory exactly horizontal initially and when he lands.

a) Assuming that his flight trajectory is a cubic polynomial of the form y=ax^3+bx^2+cx+d, deduce the relations a = -2h/l^3, b = 3h/l^2, c = d = 0.

b) If the position of his plane is (x(t),y(t), show that x(t) = l-vt, y''(t) is less than k.

c) By differentiating the flight trajectory y = ax^3+bx^2+cx+d with respect to t (not x) and using your calculations from above, deduce that Captain Sullenberger must fly with horizontal speed v satisfying 6hv^2/l^2 is less than k.

Thanks for the help!

2. We begin with the assumption that $\displaystyle (x(t),y(t))$ will travel left as $\displaystyle t$ increases. This gives us

$\displaystyle (x(0),y(0))=(l,h).$

a) In order to deduce that $\displaystyle c=d=0$, we first note that $\displaystyle (0,0)$ must lie on the trajectory, from which it follows that $\displaystyle d=0$. We then remember that when the plane lands, $\displaystyle \frac{dy}{dt}=0$, allowing us to solve a quadratic equation to prove that $\displaystyle c=0$.

b) Knowing that Captain Sullenberger flies left at speed $\displaystyle v$, we may deduce that $\displaystyle x(t)=l-vt$. Here is a hint to prove that $\displaystyle y''(t)<k$: we may use the fact that $\displaystyle a < 0$.

c) Here, we may use the Chain Rule, $\displaystyle \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}$, together with the fact that $\displaystyle \frac{dx}{dt}=-v$.

One final note: this problem is made tricky by the fact that $\displaystyle y$ can be considered either a function of $\displaystyle x$ or a function of $\displaystyle t$. The notation $\displaystyle y(0)$ is therefore ambiguous, as it could refer to the value of $\displaystyle y$ when $\displaystyle x=0$ (which is $\displaystyle 0$) or when $\displaystyle t=0$ (which is $\displaystyle h$).