IV OM - III - Task 5

A car departs from point $ O $ and drives along a straight road at a constant speed $ v $. A cyclist, who is at a distance $ a $ from point $ O $ and at a distance $ b $ from the road, wishes to deliver a letter to the car. What is the minimum speed at which the cyclist should ride to achieve his goal?
We assume that $ b > 0 $; let the Reader formulate the answer to the question in the case when $ b = 0 $, i.e., when the cyclist is on the road.
Let $ M $ be the point where the cyclist is located, $ S $ - the meeting point, $ \alpha $ - the angle $ MOS $, $ t $ - the time that will elapse from the initial moment to the moment of the meeting, and $ x $ - the speed of the cyclist. Note that






and hence, according to the Law of Sines (Fig. 48),






From this equality, it follows that $ x $ reaches its minimum value when $ \sin \beta $ is the largest. If $ \alpha < 90^\circ $, this occurs for the value $ \beta = 90^\circ $, thus






If $ \alpha \geq 90^\circ $ (Fig. 49), then the angle $ \beta $ is acute; the maximum value of $ \sin \beta $ does not exist, and the speed $ x $ is smaller the closer the angle $ \beta $ is to the acute angle $ 180^\circ - \alpha $. As the angle $ \beta $ increases and approaches $ 180^\circ - \alpha $, the speed $ x $ decreases and approaches $ v $.
Note. In the above solution, trigonometry can be avoided by reasoning as follows:
If $ \alpha < 90^\circ $ (Fig. 50) and $ MS \bot OM $, we draw $ TH \bot OM $ and $ RK \bot OM $. Then












thus, in point $ S $ on the road, the ratio of distances from points $ M $ and $ O $ is the smallest.
If $ \alpha \geq 90^\circ $ and point $ T $ is farther from point $ O $ than point $ S $ (Fig. 51), then by drawing $ NS \parallel MT $ we have






from which it is clear that the sought minimum does not exist.