\begin{problem*}{3.43}
You are standing on the ground at the origin of a coordinate system.
An airplane flies over you with constant velocity parallel to the $x$
axis at a fixed height of $7.60\E{3}\U{m}$.  At time $t=0$, the
airplane is directly above you so that the vector leading from you to
it is $\vect{P}_0=7.60\E{3}\jhat\U{m}$.  At $t=30.0\U{s}$, the
position vector leading from you to the airplane is
$\vect{P}_{30}=(8.04\E{3}\ihat+7.60\E{3}\jhat)\U{m}$ as suggested in
Figure~P3.43.  Determine the magnitude and orientation of the
airplane's position vector at $t=45.0\U{s}$.
\begin{center}
\begin{asy}
import Mechanics;

real u = 1.3cm;
real h = 1u;
real d = 8.04/7.6*h;
real dx = 0.2u;

draw((-dx,h)--(d+dx,h), dashed);
Surface s = Surface((-dx,0), (d+dx,0));
s.draw();

Vector A = Vector((0,0), mag=h, dir=90,
    L=Label("$\vect{R}_0$", position=EndPoint, align=W));
A.draw();
Vector B = Vector((0,0), mag=length((d,h)), dir=degrees((d,h)),
    L=Label("$\vect{R}_{30}$", position=EndPoint, align=RightSide));
B.draw();
\end{asy}
\end{center}
\end{problem*}

\begin{solution}
The airplane is flying at a constant velocity, so we'll compute that
velocity first,
\begin{align}
  \vect{v} &= \frac{\vect{P}_{30} - \vect{P}_0}{t_{30} - t_{0}}
    = \frac{(8.04\E{3}\ihat+7.60\E{3}\jhat)\U{m} - 7.60\E{3}\jhat\U{m}}
           {30.0\U{s} - 0\U{s}} \\
    &= \frac{8.04\E{3}\ihat}{30.0}\U{m/s}
    = 268\ihat\U{m/s}
\end{align}
Which is, as claimed in the problem text, parallel to the $x$ axis.

The position at $t=45.0\U{s}$ is therefore
\begin{align}
  \vect{P}_{45} &= \vect{P}_0 + \vect{v}(t_{45}-t_0)
    = 7.60\E{3}\jhat\U{m} + 268\ihat\U{m/s} \cdot (45.0\U{s} - 0\U{s}) \\
    &= (12.1\E{3}\ihat + 7.60\E{3}\jhat)\U{m}
    = (12.1\ihat + 7.60\jhat)\U{km} \\
  |\vect{P}_{45}| &= \sqrt{\vect{P}_{45,x}^2 + \vect{P}_{45,y}^2}
    = \sqrt{12.1^2 + 7.60^2}\U{km}
    = \ans{14.3\U{km}} \\
  \theta_{45} &= \arctan\p({\frac{\vect{P}_{45,y}}{\vect{P}_{45,x}}})
    = \arctan\p({\frac{7.60}{12.1}})
    = \ans{32.2\dg}
\end{align}
\end{solution}
