2.1. During the time it took for a slowly moving freight train to cover 1200 m, a schoolboy managed to ride his bicycle along the railway tracks from the end of the moving train to its beginning and back to the end. In doing so, the bicycle's distance meter showed that the cyclist had traveled 1800 m. Find the length of the train (in meters).
Answer: 500. Solution: Let $V$ and $U$ be the speeds of the cyclist and the train, respectively, and $h$ be the length of the train.

Then the conditions of the problem in mathematical terms can be written as follows:

$$
(V-U) t_{1}=h ; \quad(V+U) t_{2}=h ; \quad U\left(t_{1}+t_{2}\right)=l ; \quad V\left(t_{1}+t_{2}\right)=L
$$

Here $t_{1}, t_{2}$ are the times the cyclist travels in the direction of the train and against it, respectively. Taking the ratio of the last two equations, it follows that the speed of the cyclist is $a=\frac{L}{l}=1.5$ times the speed of the train. From the first two equations, we express the time and substitute it into the third. Then for the length of the train, we get the formula

$$
h=\frac{l\left(\alpha^{2}-1\right)}{2 \alpha}=500
$$