The text is in Persian. Here's its transcription and translation before analyzing the problem:

Transcription:

شناگری از ابتدای استخر به طول L یک بار در بازۀ زمانی صفر تا t۱ طول استخر را با تندی متوسط v۱ رفته و برمی‌گردد و در بازۀ زمانی t۱ تا t۲ طول استخر را با تندی متوسط v۲ طی می‌کند. در بازۀ زمانی صفر تا t۲، اندازۀ سرعت متوسط شناگر کدام است؟

Translation:

A swimmer swims from the beginning of a pool of length $L$ once during the time interval from $0$ to $t_1$, covering the length of the pool at an average speed of $v_1$ and then returns. During the interval $t_1$ to $t_2$, the swimmer covers the pool's length with an average speed of $v_2$. What is the average speed of the swimmer during the interval $0$ to $t_2$?

Solution:

To solve this problem, we need to compute the average speed over the entire time interval $0$ to $t_2$.

Step 1: Total Distance

1. From $0$ to $t_1$: The swimmer swims the pool's length $L$ back and forth, so the total distance covered is: $d_1 = 2L$
2. From $t_1$ to $t_2$: The swimmer swims the pool's length $L$ one way, so the distance covered is: $d_2 = L$
3. Total distance: $d_{\text{total}} = d_1 + d_2 = 2L + L = 3L$

Step 2: Total Time

1. From $0$ to $t_1$: The time taken is $t_1$.
2. From $t_1$ to $t_2$: The time taken is $t_2 - t_1$.
3. Total time: $t_{\text{total}} = t_1 + (t_2 - t_1) = t_2$

Step 3: Average Speed

Average speed is defined as: $v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}}$ Substitute the values: $v_{\text{avg}} = \frac{3L}{t_2}$

Final Answer:

The swimmer's average speed over the interval $0$ to $t_2$ is: $v_{\text{avg}} = \frac{3L}{t_2}$

Do you need further clarifications or detailed examples?

Here are 5 related questions for practice:

1. How would the average speed change if $v_2$ were double $v_1$?
2. What happens to the average speed if $t_1 = t_2$?
3. How is the calculation affected if the swimmer's return distance had a different speed?
4. How can we calculate the swimmer's average velocity instead of average speed?
5. If $L = 50$ meters, $t_1 = 20$ seconds, and $t_2 = 40$ seconds, what is the numerical value of $v_{\text{avg}}$?

Tip: Average speed depends only on total distance and total time, not the speeds during individual intervals.