Solution

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

$$L = s \cdot \cos\frac{\alpha}{2};\quad h = s \cdot\sin\frac{\alpha}{2}$$

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

$$\fbox{$\alpha = 2 \cdot \arctan \frac{u}{v}$}$$