Let's solve the average velocity problems (numbers 1-10) from the document, assuming a position-time graph is provided for a particle moving along the \(x\)-axis. Since the graph details are not visible in the provided file, I'll walk through the general approach and you can plug in the specific numbers from the graph.

### Average Velocity Calculations:

**Average Velocity Formula:**
\[
v_{\text{avg}} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}
\]

### Solutions:

1. **(a) From 0 to 2 s:**
   \[
   v_{\text{avg}} = \frac{x(2 \text{ s}) - x(0 \text{ s})}{2 \text{ s} - 0 \text{ s}} = \frac{x_2 - x_0}{2}
   \]

2. **(b) From 0 to 4 s:**
   \[
   v_{\text{avg}} = \frac{x(4 \text{ s}) - x(0 \text{ s})}{4 \text{ s} - 0 \text{ s}} = \frac{x_4 - x_0}{4}
   \]

3. **(c) From 2 to 4 s:**
   \[
   v_{\text{avg}} = \frac{x(4 \text{ s}) - x(2 \text{ s})}{4 \text{ s} - 2 \text{ s}} = \frac{x_4 - x_2}{2}
   \]

4. **(d) From 4 to 7 s:**
   \[
   v_{\text{avg}} = \frac{x(7 \text{ s}) - x(4 \text{ s})}{7 \text{ s} - 4 \text{ s}} = \frac{x_7 - x_4}{3}
   \]

5. **(e) From 0 to 8 s:**
   \[
   v_{\text{avg}} = \frac{x(8 \text{ s}) - x(0 \text{ s})}{8 \text{ s} - 0 \text{ s}} = \frac{x_8 - x_0}{8}
   \]

### Example Calculation:

Let's assume the following positions based on the graph (these values are hypothetical as the graph isn't visible):

- \( x(0 \text{ s}) = 0 \) m
- \( x(2 \text{ s}) = 4 \) m
- \( x(4 \text{ s}) = 8 \) m
- \( x(7 \text{ s}) = 14 \) m
- \( x(8 \text{ s}) = 16 \) m

Substituting into the formulas:

1. **(a) From 0 to 2 s:**
   \[
   v_{\text{avg}} = \frac{4 \text{ m} - 0 \text{ m}}{2 \text{ s}} = \frac{4}{2} = 2 \text{ m/s}
   \]

2. **(b) From 0 to 4 s:**
   \[
   v_{\text{avg}} = \frac{8 \text{ m} - 0 \text{ m}}{4 \text{ s}} = \frac{8}{4} = 2 \text{ m/s}
   \]

3. **(c) From 2 to 4 s:**
   \[
   v_{\text{avg}} = \frac{8 \text{ m} - 4 \text{ m}}{2 \text{ s}} = \frac{4}{2} = 2 \text{ m/s}
   \]

4. **(d) From 4 to 7 s:**
   \[
   v_{\text{avg}} = \frac{14 \text{ m} - 8 \text{ m}}{3 \text{ s}} = \frac{6}{3} = 2 \text{ m/s}
   \]

5. **(e) From 0 to 8 s:**
   \[
   v_{\text{avg}} = \frac{16 \text{ m} - 0 \text{ m}}{8 \text{ s}} = \frac{16}{8} = 2 \text{ m/s}
   \]

### Summary:
Based on the assumed position values, the average velocity in all these intervals would be \( 2 \text{ m/s} \). 

If you have the specific position values from the graph, you can substitute them into the formulas to get the exact values for each interval.

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Do you need help with more specific numbers, or is there another part you'd like to explore further?

### Relative Questions:
1. How do you interpret the average velocity when it is constant across different intervals?
2. What would be the average velocity if the particle returns to its starting position at any point?
3. How would you calculate the average speed if the particle moves in both positive and negative directions?
4. Can the average velocity be zero even if the particle has moved? How?
5. How do you graphically determine average velocity from a position-time graph?

### Tip:
Always ensure that your time intervals are consistent and clearly defined when calculating average velocities, especially when working with piecewise graphs.

Learn how to calculate average velocity for different time intervals using a position-time graph. Explore examples and formulas for intervals from 0 to 2 seconds, 0 to 4 seconds, 2 to 4 seconds, 4 to 7 seconds, and 0 to 8 seconds. Understand the method to derive average velocity when specific position values are known.

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