## Task 23/61

On a $22.5 \mathrm{~km}$ long tram line, during the time from $8^{h}$ to $16^{h}$, the trains should run in both directions at 10-minute intervals. The first trains of this operating time leave the end stations at $8^{h}$. Their average speed (including stop times) is $18 \frac{km}{h}$. The train personnel should have a break of at least 10 and at most 20 minutes at the end stations.

1. When does the first train departing from end station A leave this end station for the second time?
2. How many trains must be used on this line during the operating time from $8^{h}$ to $16^{h}$? Trains that are still on the line from the morning traffic before $8^{h}$ and are being taken out, as well as trains that go on the line in advance for the morning traffic starting at $16^{h}$, should not be included.
3. At what time intervals do the trains meet each other?
1. From the formula $s=v \cdot t$ (where $s$ is the distance, $v$ is the speed, and $t$ is the time), it follows that $t=\frac{s}{v}$. In our case, $s=22.5 \mathrm{~km}$ and $v=18 \frac{\mathrm{km}}{\mathrm{h}}$.

Thus, $t=\frac{22.5}{18} \mathrm{~h}=1 \mathrm{~h} 15 \mathrm{~min}$.

The first train arrives at the final stop B at $9^{h} 15^{\min}$. Due to the prescribed break, it departs at $9^{h} 30^{\min}$. Since the same conditions apply for the return trip, it begins its second journey from the final stop A at $11^{h} 00^{\text {min }}$.

2. The first train departing from A starts its return trip from B 90 minutes after its departure. During this time, nine trains have left from both A and B, making a total of 18 trains.
3. The time intervals are 5 minutes. This result can be obtained from several considerations, for example, the following:

We consider the encounters of a train a, traveling from A to B, with two consecutive trains $b_{1}$ and $b_{2}$, traveling from B to A. At the time of the encounter between a and $b_{1}$, $b_{2}$ is 10 minutes away from the meeting point of a and $b_{1}$.

Since a and $b_{2}$ have the same speed, each of these two trains must cover half the distance corresponding to this travel gap to reach their meeting point. This takes 5 minutes.

## Solution by Walter Rulff:

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-023.jpg?height=417&width=1591&top_left_y=268&top_left_x=241)

Graphical Solution:

We represent the journey of the trams in a right-angled time-distance diagram (graphical timetable - figure). From the given speed $v=18 \frac{\mathrm{km}}{\mathrm{h}}$, it follows that each tram is 18 km away from the departure point one hour after departure. This defines the time-distance curves (at least approximately, as stop times and speed fluctuations are not considered).

The results can be read directly from the diagram.