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{Exam 3 \hfill Analysis in Mechanical Engineering \hfill 11/16/93 }\\
{\it Closed book \hfill Van Dommelen \hfill 2:45-4:00}
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Show all reasoning. One book of mathematical tables may be used.

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\item
The direction of the field lines of a two-dimensional
electrostatic field is given by:
$$E_y {\rm d} x - E_x {\rm d} y = 0$$
Find an algebraic expression for the field lines of the field:
$$E_x = e^x \cos (y) + x - y$$
$$E_y = -e^x \sin (y) - x - y$$

\item
For the RCL circuit shown below, the voltage across the condensator
satisfies
$${{\rm d}^2 V\over {\rm d} t^2} + {R\over L} {{\rm d} V\over {\rm d} t}
+ {1\over LC} V = e(t)$$
Assuming that the resistance $R=3$, capacitance $C=0.5$ and inductance
$L=1$, and the applied voltage is $e(t)= - e^{-t}/(1+e^t)$,
find the voltage across the condensator.

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\item
A projectile is fired through a constant area, but variable
density duct.  Assuming the velocity variation is linear with distance,
the equation of motion for the projectile is
$$ \ddot x = a {(x - \dot x)^2 \over x}$$
where $a$ is a constant that cannot be scaled out.
Assuming that $a=1$, find an algebraic expression relating the projectile
velocity $\dot x$ to its position $x$.
Verify that the projectile never catches up with the stream:
projectile velocity does not become equal to position at large $x$.

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