\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\begin{document}
\parindent=0pt

{\bf Question}

In steady rain, raindrops fall at 16kmh$^{-1}$.  Draw up a table
of values of the angles of raindrop streaks on train windows which
would enable a passenger in the train to estimate his speed in
units of 10kmh$^{-1}$ up to 100kmh$^{-1}$

\vspace{.25in}

{\bf Answer}

We require to find the velocity of raindrops relative to the
train, ${\bf v}_{DT}$.

So we will use the equation $\ds {\bf v}_{DT} = {\bf v}_D - {\bf
v}_T$, where ${\bf v}_D$ is the velocity of the raindrops and
${\bf v}_T$ is the velocity of the train.

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\put(1,.2){\circle{.4}}

\put(1.8,.2){\circle{.4}}

\put(3.5,.2){\circle{.4}}

\put(4.5,.2){\circle{.4}} %wheels%

\put(0.7,0.4){\line(1,0){1.5}}

\put(0.7,0.4){\line(0,1){.75}}

\put(2.2,0.4){\line(0,1){.75}}

\put(0.7,1.15){\line(1,0){1.5}} %carriage%

\put(3.,0.4){\line(1,0){2}}

\put(3,0.4){\line(0,1){0.75}}

\put(5,0.4){\line(0,1){0.5}}

\put(3,1.15){\line(1,0){1.73}}

\put(5,0.9){\line(-1,1){.28}}

\put(5,0.9){\line(-1,0){.35}}

\put(4.65,0.9){\line(0,1){.27}} %locomotive%

\put(2.2,.7){\line(1,0){.8}}%line connecting the two trains%

\put(5.5,0.8){\vector(1,0){1}}

\put(6.6,0.8){\makebox(0,0)[l]{${\bf v}_T = v_T{\bf i}$}}

\put(0.5,1.5){\line(0,-1){0.75}}

\put(0.5,1.7){\vector(0,-1){0.25}}

\put(0.6,1.5){\makebox(0,0)[l]{${\bf v}_D = -16{\bf j}$}}

\put(0,0){\vector(0,1){.3}}

\put(0,0){\vector(1,0){.3}}

\put(0,0.6){\makebox(0,0)[l]{\bf j}}

\put(0.33,0){\makebox(0,0)[l]{\bf i}}

\end{picture}
\end{center}

\bigskip


$\ds{\bf v}_{DT} = -16{\bf j} - {\bf v}_T{\bf i} \Rightarrow
\theta = \tan^{-1}\frac{16}{{\bf v}_T}$

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\put(2,2.5){\vector(0,-1){0.75}}

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\put(2,1){\vector(-1,0){1}}

\put(0,1){\line(4,3){2}}

\put(2,2.5){\vector(-4,-3){1}}

\put(0.5,1.1){$\theta$}

\put(2,1.2){\line(-1,0){.2}}

\put(1.8,1){\line(0,1){.2}}

\put(0.7,0.7){$-v_T{\bf i}$}

\put(2.2,1.7){$-16${\bf j}}

\put(0.5,2){${\rm \bf v}_{DT}$}
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\begin{center}
$\begin{array}{cccccc} {\bf v}_T\ \  & \theta (rads)\ \  &{\bf
v}_T \ \ & \theta (rads)\ \  &{\bf v}_T\ \  & \theta (rads)
\\ 0 & 0 & 40 & 0.38 & 80 & 0.197 \\
 10 &1.01 & 50 & 0.31 & 90 & 0.176 \\
 20 & 0.67 & 60  & 0.26 & 100 & 0.158 \\ 30 & 0.48 & 70 & 0.22\\
\end{array}$
\end{center}



\end{document}