$1.1.10^*.$ 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 $u = v/2$.
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: $AC = L, BC = h, AB = s.$
2. From right-angled triangle $ABC$ we have
$$L = s \cdot cos \frac{\alpha}{2}\text{ и }h = s \cdot sin \frac{\alpha}{2}$$
3. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$
${t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha/2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha/2)}{{u}}$
where from
$$\fbox{$\alpha = 2 \cdot \text{arcsin} \frac{u}{v}$}$$
From the region bounded by the angle $α = 2 \, \text{arcsin}(u/v)$ with vertex at the point $A$, bisected by the motorway