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{\bf Question}

An object falls under gravity near the surface of the earth, and
its motion is impeded by air resistance proportional to its speed.
The equation of motion for the object is the second order
differential equation

$$\frac{d^2y}{dx^2} = -g-k\frac{dy}{dx}$$

where $k$ and $g$ are positive constants.

\begin{description}
\item[(i)]
Show by letting $v=\frac{dy}{dt}$ we can obtain the first order
differential equation

$$\frac{dv}{dt} + kt = -g$$

and hence using an integrating factor find the velocity of the
object as a function of time, given that at $t = 0$ it was
initially at speed $v = v_0$.
\item[(ii)]
Find the limiting velocity $lim_{t \rightarrow \infty}v(t)$.
\item[(iii)]
If the object was initally at height $y_0$, find its height $y(t)$
at time $t$.
\end{description}


{\bf Answer}



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