| ### Statement | | ### Statement |
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| $1.1.10^*.$ 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 $u = v/2$. | | $1.1.10^*.$ 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 $u = v/2$. |
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| ### Solution | | ### Solution |
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| 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: $AC = L, BC = h, AB = s.$ | | 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: $AC = L, BC = h, AB = s.$ |
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| 2\. From right-angled triangle $ABC$ we have | | 2\. From right-angled triangle $ABC$ we have |
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| $$ | | $$ |
| L = s \cdot \cos\frac{\alpha}{2};\quad h = s \cdot\sin\frac{\alpha}{2} | | L = s \cdot \cos\frac{\alpha}{2};\quad h = s \cdot\sin\frac{\alpha}{2} |
| $$ | | $$ |
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| 3\. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$ | | 3\. Travel time of bus $t_1$ and passenger $t_2$ before meeting at point $C$ |
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| {t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha /2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha /2)}{{u}} | | {t}_{1}=\frac{{L}}{{v}}=\frac{{s}\cos(\alpha /2)}{{v}};\quad{t}_{2}=\frac{{h}}{{u}}=\frac{{s}\sin(\alpha /2)}{{u}} |
| $$ | | $$ |
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| where from | | where from |
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| \fbox{$\alpha = 2 \cdot \arctan \frac{u}{v}$} | | \fbox{$\alpha = 2 \cdot \arctan \frac{u}{v}$} |
| $$ | | $$ |