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related_rates.html
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related_rates.html
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<h1>Related rates</h1>
<p>Related rates problems involve two (or more) unknown quantities that are related through an equation. As the two variables depend on each other, also so do their rates–-change with respect to some variable which is often time, though exactly how remains to be discovered. Hence the name "related rates."</p>
<h2>Examples</h2>
<p>The following is a typical "book" problem:</p>
<blockquote>
<p>A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is expanding at the rate of 4 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when its dimensions are 12 cm by 8 cm? <a href="http://oregonstate.edu/instruct/mth251/cq/Stage9/Practice/ratesProblems.html">Source.</a></p>
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"/>
<figcaption><div class="markdown"><p>As $t$ increases, the size of the rectangle grows. The ratio of width to height is fixed. If we know the rate of change in time for the width ($dw/dt$) and the height ($dh/dt$) can we tell the rate of change of <em>area</em> with respect to time ($dA/dt$)?</p>
</div></figcaption>
</figure>
</div>
<p>Here we know <span class="math">$A = w \cdot h$</span> and we know some things about how <span class="math">$w$</span> and <span class="math">$h$</span> are related <em>and</em> about the rate of how both <span class="math">$w$</span> and <span class="math">$h$</span> grow in time <span class="math">$t$</span>. That means that we could express this growth in terms of some functions <span class="math">$w(t)$</span> and <span class="math">$h(t)$</span>, then we can figure out that the area–-as a function of <span class="math">$t$</span>–-will be expressed as:</p>
<p class="math">\[
~
A(t) = w(t) \cdot h(t).
~
\]</p>
<p>We would get by the product rule that the <em>rate of change</em> of area with respect to time, <span class="math">$A'(t)$</span> is just:</p>
<p class="math">\[
~
A'(t) = w'(t) h(t) + w(t) h'(t).
~
\]</p>
<p>As an aside, it is fairly conventional to suppress the <span class="math">$(t)$</span> part of the notation <span class="math">$A=wh$</span> and to use the Leibniz notation for derivatives:</p>
<p class="math">\[
~
\frac{dA}{dt} = \frac{dw}{dt} h + w \frac{dh}{dt}.
~
\]</p>
<p>This relationship is true for all <span class="math">$t$</span>, but the problem discusses a certain value of <span class="math">$t$</span>–-when <span class="math">$w(t)=8$</span> and <span class="math">$h(t) = 12$</span>. At this same value of <span class="math">$t$</span>, we have <span class="math">$w'(t) = 4$</span> and so <span class="math">$h'(t) = 6$</span>. Substituting these 4 values into the 4 unknowns in the formula for <span class="math">$A'(t)$</span> gives:</p>
<p class="math">\[
~
A'(t) = 4 \cdot 12 + 8 \cdot 6 = 96.
~
\]</p>
<p>Summarizing, from the relationship between <span class="math">$A$</span>, <span class="math">$w$</span> and <span class="math">$t$</span>, there is a relationship between their rates of growth with respect to <span class="math">$t$</span>, a time variable. Using this and known values, we can compute. In this case <span class="math">$A'$</span> at the specific <span class="math">$t$</span>.</p>
<p>We could also have done this differently. We would recognize the following:</p>
<ul>
<li><p>The area of a rectangle is just:</p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-n'>w</span><span class='hljl-p'>,</span><span class='hljl-n'>h</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>w</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>h</span>
</pre>
<pre class="output">
A (generic function with 1 method)
</pre>
<ul>
<li><p>The width–-expanding at a rate of <span class="math">$4t$</span> from a starting value of <span class="math">$2$</span>–-must satisfy:</p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-nf'>w</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-ni'>4</span><span class='hljl-oB'>*</span><span class='hljl-n'>t</span>
</pre>
<pre class="output">
w (generic function with 1 method)
</pre>
<ul>
<li><p>The height is a constant proportion of the width:</p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-nf'>h</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>3</span><span class='hljl-oB'>/</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-nf'>w</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
h (generic function with 1 method)
</pre>
<p>This means again that area depends on <span class="math">$t$</span> through this formula:</p>
<pre class='hljl'>
<span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-nf'>w</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-nf'>h</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span>
</pre>
<pre class="output">
A (generic function with 2 methods)
</pre>
<p>This is why the rates of change are related: as <span class="math">$w$</span> and <span class="math">$h$</span> change in time, the functional relationship with <span class="math">$A$</span> means <span class="math">$A$</span> also changes in time.</p>
<p>Now to answer the question, when the width is 8, we must have that <span class="math">$t$</span> is:</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CalculusWithJulia</span><span class='hljl-t'> </span><span class='hljl-cs'># loads `Plots`, `ForwardDiff`, `Roots`</span><span class='hljl-t'>
</span><span class='hljl-n'>tstar</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>fzero</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>-></span><span class='hljl-t'> </span><span class='hljl-nf'>w</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-ni'>8</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>4</span><span class='hljl-p'>])</span><span class='hljl-t'> </span><span class='hljl-cs'># or solve by hand to get 3/2</span>
</pre>
<pre class="output">
1.5
</pre>
<p>The question is to find the rate the area is increasing at the given time <span class="math">$t$</span>, which is <span class="math">$A'(t)$</span> or <span class="math">$dA/dt$</span>. We get this by performing the differentiation, the substituting in the value.</p>
<p>Here we do so with the aid of <code>Julia</code>, though this problem could readily be done "by hand."</p>
<p>We (again) re-express <span class="math">$A$</span> as a function of <span class="math">$t$</span> by composition, then differentiate that:</p>
<pre class='hljl'>
<span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-nf'>w</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-nf'>h</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-n'>da_dt</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>A</span><span class='hljl-oB'>'</span><span class='hljl-p'>(</span><span class='hljl-n'>tstar</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
96.0
</pre>
<p>So what? Why is 96 of any interest? It is if the value at a specific time is needed. But in general, a better question might be to understand if there is some pattern to the numbers in the figure, these being <span class="math">$6, 54, 150, 294, 486, 726$</span>. Their differences are the <em>average</em> rate of change:</p>
<pre class='hljl'>
<span class='hljl-n'>xs</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-ni'>6</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>54</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>150</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>294</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>486</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>726</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>ds</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-n'>xs</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
5-element Array{Int64,1}:
48
96
144
192
240
</pre>
<p>Those seem to be increasing by a fixed amount each time, which we can see by one more application of <code>diff</code>:</p>
<pre class='hljl'>
<span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-n'>ds</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
4-element Array{Int64,1}:
48
48
48
48
</pre>
<p>How can this relationship be summarized? Well, let's go back to what we know, though this time using symbolic math:</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>SymPy</span><span class='hljl-t'>
</span><span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-t'>
</span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-nf'>A</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}48.0 t + 24.0\end{equation*}</div>
<p>This should be clear: the rate of change, <span class="math">$dA/dt$</span>, is increasing linearly, hence the second derivative, <span class="math">$dA^2/dt^2$</span> would be constant, just as we saw for the average rate of change.</p>
<p>So, for this problem, a constant rate of change in width and height leads to a linear rate of change in area, put otherwise, linear growth in both width and height leads to quadratic growth in area.</p>
<h5>Example</h5>
<p>A ladder, with length <span class="math">$l$</span>, is leaning against a wall. We parameterize this problem so that the top of the ladder is at <span class="math">$(0,h)$</span> and the bottom at <span class="math">$(b, 0)$</span>. Then <span class="math">$l^2 = h^2 + b^2$</span> is a constant.</p>
<p>If the ladder starts to slip away at the base, but remains in contact with the wall, express the rate of change of <span class="math">$h$</span> with respect to <span class="math">$t$</span> in terms of <span class="math">$db/dt$</span>.</p>
<p>We have from implicitly differentiating in <span class="math">$t$</span> the equation <span class="math">$l^2 = h^2 + b^2$</span>, noting that <span class="math">$l$</span> is a constant, that:</p>
<p class="math">\[
~
0 = 2h \frac{dh}{dt} + 2b \frac{db}{dt}.
~
\]</p>
<p>Solving, yields:</p>
<p class="math">\[
~
\frac{dh}{dt} = -\frac{b}{h} \cdot \frac{db}{dt}.
~
\]</p>
<ul>
<li><p>If <span class="math">$l = 12$</span> and <span class="math">$db/dt = 2$</span> when <span class="math">$b=4$</span>, find <span class="math">$dh/dt$</span>.</p>
</li>
</ul>
<p>We just need to find <span class="math">$h$</span> for this value of <span class="math">$b$</span>, as the other two quantities in the last equation are known. But <span class="math">$h = \sqrt{l^2 - b^2}$</span>, so the answer is:</p>
<pre class='hljl'>
<span class='hljl-n'>l</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>dbdt</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>12</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>4</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>2</span><span class='hljl-t'>
</span><span class='hljl-n'>height</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>sqrt</span><span class='hljl-p'>(</span><span class='hljl-n'>l</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-n'>b</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-oB'>-</span><span class='hljl-n'>b</span><span class='hljl-oB'>/</span><span class='hljl-n'>height</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>dbdt</span>
</pre>
<pre class="output">
-0.7071067811865475
</pre>
<ul>
<li><p>What happens to the rate as <span class="math">$b$</span> goes to <span class="math">$l$</span>?</p>
</li>
</ul>
<p>As <span class="math">$b$</span> goes to <span class="math">$l$</span>, <span class="math">$h$</span> goes to 0, so <span class="math">$b/h$</span> blows up. Unless <span class="math">$db/dt$</span> goes to <span class="math">$0$</span>, the expression will become <span class="math">$-\infty$</span>.</p>
<h5>Example</h5>
<div class="well well-sm">
<figure>
<img src="data:image/gif;base64,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"/>
<figcaption><div class="markdown"><p>The flight of the ball as being tracked by a stationary outfielder. This ball will go over the head of the player. What can the player tell from the quantity $d\theta/dt$?</p>
</div></figcaption>
</figure>
</div>
<p>A baseball player stands 100 meters from home base. A batter hits the ball directly at the player so that the distance from home plate is <span class="math">$x(t)$</span> and the height is <span class="math">$y(t)$</span>.</p>
<p>The player tracks the flight of the ball in terms of the angle <span class="math">$\theta$</span> made between the ball and the player. This will satisfy:</p>
<p class="math">\[
~
\tan(\theta) = \frac{y(t)}{100 - x(t)}.
~
\]</p>
<p>What is the rate of change of <span class="math">$\theta$</span> with respect to <span class="math">$t$</span> in terms of that of <span class="math">$x$</span> and <span class="math">$y$</span>?</p>
<p>We have by the chain rule and quotient rule:</p>
<p class="math">\[
~
\sec^2(\theta) \theta'(t) = \frac{y'(t) \cdot (100 - x(t)) - y(t) \cdot (-x'(t))}{(100 - x(t))^2}.
~
\]</p>
<p>If we have <span class="math">$x(t) = 50t$</span> and <span class="math">$y(t)=v_{0y} t - 5 t^2$</span> when is the rate of change of the angle happening most quickly?</p>
<p>The formula for <span class="math">$\theta'(t)$</span> is</p>
<p class="math">\[
~
\theta'(t) = \cos^2(\theta) \cdot \frac{y'(t) \cdot (100 - x(t)) - y(t) \cdot (-x'(t))}{(100 - x(t))^2}.
~
\]</p>
<p>This question requires us to differentiate <em>again</em> in <span class="math">$t$</span>. Since we have fairly explicit function for <span class="math">$x$</span> and <span class="math">$y$</span>, we will use <code>SymPy</code> to do this.</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-t'>
</span><span class='hljl-n'>theta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>SymFunction</span><span class='hljl-p'>(</span><span class='hljl-s'>"theta"</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>v0</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>5</span><span class='hljl-t'>
</span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>50</span><span class='hljl-n'>t</span><span class='hljl-t'>
</span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>v0</span><span class='hljl-oB'>*</span><span class='hljl-n'>t</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-ni'>5</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'>
</span><span class='hljl-n'>eqn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>tan</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>/</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span>
</pre>
<div class="well well-sm">\begin{equation*}\tan{\left (\theta{\left (t \right )} \right )} - \frac{- 5 t^{2} + 5 t}{- 50 t + 100}\end{equation*}</div>
<pre class='hljl'>
<span class='hljl-n'>thetap</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>dtheta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>solve</span><span class='hljl-p'>(</span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-n'>eqn</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>thetap</span><span class='hljl-p'>)[</span><span class='hljl-ni'>1</span><span class='hljl-p'>]</span>
</pre>
<div class="well well-sm">\begin{equation*}\frac{\left(- t^{3} + 3 t^{2} + 2 t \left(t - 2\right)^{2} - 2 t - \left(t - 2\right)^{2}\right) \cos^{2}{\left (\theta{\left (t \right )} \right )}}{10 \left(t - 2\right)^{3}}\end{equation*}</div>
<p>We could proceed directly by evaluating:</p>
<pre class='hljl'>
<span class='hljl-n'>d2theta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-n'>dtheta</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-p'>)(</span><span class='hljl-n'>thetap</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-n'>dtheta</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}\frac{\left(- 3 t^{2} + 2 t \left(2 t - 4\right) + 4 t + 2 \left(t - 2\right)^{2} + 2\right) \cos^{2}{\left (\theta{\left (t \right )} \right )}}{10 \left(t - 2\right)^{3}} - \frac{3 \left(- t^{3} + 3 t^{2} + 2 t \left(t - 2\right)^{2} - 2 t - \left(t - 2\right)^{2}\right) \cos^{2}{\left (\theta{\left (t \right )} \right )}}{10 \left(t - 2\right)^{4}} - \frac{\left(- t^{3} + 3 t^{2} + 2 t \left(t - 2\right)^{2} - 2 t - \left(t - 2\right)^{2}\right)^{2} \sin{\left (\theta{\left (t \right )} \right )} \cos^{3}{\left (\theta{\left (t \right )} \right )}}{50 \left(t - 2\right)^{6}}\end{equation*}</div>
<p>That is not so tractable, however.</p>
<p>It helps to simplify <span class="math">$\cos^2(\theta(t))$</span> using basic right-triangle trigonometry. Recall, <span class="math">$\theta$</span> comes from a right triangle with height <span class="math">$y(t)$</span> and length <span class="math">$(100 - x(t))$</span>. The cosine of this angle will be <span class="math">$100 - x(t)$</span> divided by the length of the hypotenuse. So we can substitute:</p>
<pre class='hljl'>
<span class='hljl-n'>dtheta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>dtheta</span><span class='hljl-p'>(</span><span class='hljl-nf'>cos</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-oB'>/</span><span class='hljl-p'>(</span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-oB'>-</span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>))</span>
</pre>
<div class="well well-sm">\begin{equation*}\frac{\left(- 50 t + 100\right)^{2} \left(- t^{3} + 3 t^{2} + 2 t \left(t - 2\right)^{2} - 2 t - \left(t - 2\right)^{2}\right)}{10 \left(t - 2\right)^{3} \left(\left(- 50 t + 100\right)^{2} + \left(- 5 t^{2} + 5 t\right)^{2}\right)}\end{equation*}</div>
<p>Plotting reveals some interesting things. For <span class="math">$v_{0y} < 10$</span> we have graphs that look like:</p>
<pre class='hljl'>
<span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>dtheta</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>v0</span><span class='hljl-oB'>/</span><span class='hljl-ni'>5</span><span class='hljl-p'>)</span>
</pre>
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<p>The ball will drop in front of the player, and the change in <span class="math">$d\theta/dt$</span> is monotonic.</p>
<p>But let's rerun the code with <span class="math">$v_{0y} > 10$</span>:</p>
<pre class='hljl'>
<span class='hljl-n'>v0</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>15</span><span class='hljl-t'>
</span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>50</span><span class='hljl-n'>t</span><span class='hljl-t'>
</span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>v0</span><span class='hljl-oB'>*</span><span class='hljl-n'>t</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-ni'>5</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'>
</span><span class='hljl-n'>eqn</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>tan</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>/</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-n'>thetap</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>dtheta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>solve</span><span class='hljl-p'>(</span><span class='hljl-nf'>diff</span><span class='hljl-p'>(</span><span class='hljl-n'>eqn</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>t</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>thetap</span><span class='hljl-p'>)[</span><span class='hljl-ni'>1</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>dtheta</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>subs</span><span class='hljl-p'>(</span><span class='hljl-n'>dtheta</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-nf'>cos</span><span class='hljl-p'>(</span><span class='hljl-nf'>theta</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-oB'>/</span><span class='hljl-p'>(</span><span class='hljl-nf'>y</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>100</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>x</span><span class='hljl-p'>(</span><span class='hljl-n'>t</span><span class='hljl-p'>))</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>dtheta</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>v0</span><span class='hljl-oB'>/</span><span class='hljl-ni'>5</span><span class='hljl-p'>)</span>
</pre>
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