\input epsf
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\noindent {\bf Problem statement} A freely falling body starting from
rest has velocity $v= gt$ and displacement 
${s= {1\over 2} gt^2}$ where $t$ is the time elapsed since
rest.  Suppose the freely falling body starts at rest and falls
1,000 feet.

\medskip

\noindent a) Calculate the time $T$ (in seconds) this takes (here $g =
32$ ft/s$^2$) and the {\it time average}\/ of the velocity of the
body: $ v_{\rm time\ aver}={1\over T}\int_0^T v(t)\, dt.$ Draw a graph
of the function $v(t)$ for $0\leq t \leq T$. Find the time $t$ when
$v(t) = v_{\rm time\ aver}$ and give a graphical interpretation.

\medskip

\noindent b) Find a formula for the velocity as a function $f(s)$ of
displacement $s$, and calculate the {\it distance average}\/ of the
velocity: $ v_{\rm dist\ aver} = {1\over {1000}} \int_0^{1000} f(s) \,
ds.$ Draw a graph of the function $v=f(s)$ for $0\leq s \leq
1000$. Find the distance $s$ that the body has fallen when $f(s) =
v_{\rm dist\ aver}$ and give a graphical interpretation.

\medskip

\noindent {\bf Note} $v_{\rm dist\ aver} \neq v_{\rm time
\ aver}\,$! Every user of statistics (this means, essentially, every
person in this course) should do this problem.
Averages can be difficult to understand.










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