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\noindent {\bf Question}
\begin{enumerate}
\item A ball has {\bf bounce coefficient} $0<r<1$ if, when it is
dropped from height $h$, it bounces back to a height of $rh$.
Suppose that such a ball is dropped from the initial height $a$
and subsequently bounces infinitely many times.  Determine the
total up-and-down distance the ball travels.
\item Two cars, driven by Jack and Jill, are begin driven towards each
other, with Jack driving at 25 miles per hour and Jill driving at
95 miles per hour.  When the cars are 120 miles apart, a fly
leaves the front of Jack's car and flies to Jill's car at 257
miles per hour; when it reaches Jill's car, it immediately turns
around and flies back to Jack's car, and keeps going back and
forth until it is crushed between the two cars when they crash
together.  Assuming the fly loses no time in changing direction,
calculate the total distance the fly has flown in its journey
between the two cars.
\end{enumerate}
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\noindent {\bf Answer}
\begin{enumerate}
\item Before hitting the ground the first time, the ball travels
distance $a$.  Between hitting the ground the first and second
times, the ball travels distance $2ra$ (distance $ra$ up from the
ground, and then distance $ra$ back to down to earth again).
Between hitting the ground the second and third times, the ball
travels distance $2r^2a$ (distance $r^2a$ up from the ground, and
then distance $r^2a$ back to down to earth again).   Between
hitting the ground the $n^{th}$ and the $(n+1)^{st}$ times, the
ball travels distance $2r^na$ (distance $r^na$ up from the ground,
and then distance $r^na$ back to down to earth again).  Hence, the
total distance travelled is
\[ a + 2ra + 2 r^2 a + \ldots =a +\sum_{n=1}^\infty 2r^n a = a +
2ra \sum_{n=1}^\infty r^{n-1} =a + 2ra\sum_{k=0}^\infty r^k\]

$ = a + \frac{2ra}{1-r} = \frac{a + ra}{1-r}.$

\item One way to do this problem is to actually write out the
appropriate geometric series and summing it.  The easier way is to
note that the cars will crash exactly one hour after the fly
leaves the front of Jack's car, and in that hour (given the
assumption that the fly loses no time in changing direction) the
fly flies exactly 257 miles.
\end{enumerate}

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