Savchenko Solutions
<h3 id="back-link"><a href="/#1.1">$\leftarrow$Back</a></h3>
<h3> Statement </h3>
<p>
$1.1.10^*.$ A bus is driving along a straight highway at constant speed $v$. You have noticed the bus when it was at some point $A$. From what area near the highway can you catch up with this bus if your running speed is $u < v$? Draw this area for $u = v/2$.
For problem $1.1.10^*$
<h3>Solution</h3>
<p>
<center>
<figure>
<img src="1.1.10.png" alt="1.1.10"
loading="lazy" width="200" />
<figcaption>
Chasing a bus
</figcaption>
</figure>
</center>
<p>
1. Let the bus is at point $A$, and the catching up person starts from point $B$ and runs perpendicular to the roadway $AC$. Let us introduce the notations: $AC = L, BC = h, AB = s.$
</p>
<center>
<figure>
<img src="sol.png" alt="1.1.10"
loading="lazy" width="200" />
<figcaption>
The plane bounded by $\alpha$
</figcaption>
</figure>
</center>
<p>
2. From right-angled triangle $ABC$ we have
</p>
<p style="text-align: center;">
$$L = s \cdot cos \frac{\alpha}{2}\text{ и }h = s \cdot sin \frac{\alpha}{2}$$
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<p>
3. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$
</p>
<p style="text-align: center;">
${t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha/2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha/2)}{{u}}$
</p>
<p>
where from
</p>
<p style="text-align: center;">
$$\fbox{$\alpha = 2 \cdot \text{arcsin} \frac{u}{v}$}$$
</p>
</p>
<h4>Answer</h4>
<p>
From the region bounded by the angle $α = 2 \\, \text{arcsin}(u/v)$ with vertex at the point $A$, bisected by the motorway
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