https://file.aiquickdraw.com/mathbot/file/17240501836390r319fih.jpg

solve the problem that is shown on the image and give the solutions

To solve the problem presented in the image, we will address each of the questions based on the provided \(x\)-\(t\) graph (position vs. time graph).

### Part A: Create the \(v\)-\(t\) graph (velocity vs. time graph)

The slope of the \(x\)-\(t\) graph gives us the velocity. To create the \(v\)-\(t\) graph, we need to find the slopes of each segment in the \(x\)-\(t\) graph.

1. **First segment (from \( t = 0 \) to \( t = 3 \) seconds)**:
   - The line rises from 0 to 2. 
   - Slope (velocity) = \(\frac{2 - 0}{3 - 0} = \frac{2}{3} \, \text{m/s}\).

2. **Second segment (from \( t = 3 \) to \( t = 5 \) seconds)**:
   - The line drops from 2 to 0.
   - Slope (velocity) = \(\frac{0 - 2}{5 - 3} = \frac{-2}{2} = -1 \, \text{m/s}\).

3. **Third segment (from \( t = 5 \) to \( t = 7 \) seconds)**:
   - The line rises from 0 to 2.
   - Slope (velocity) = \(\frac{2 - 0}{7 - 5} = \frac{2}{2} = 1 \, \text{m/s}\).

4. **Fourth segment (from \( t = 7 \) to \( t = 8 \) seconds)**:
   - The line drops from 2 to -1.
   - Slope (velocity) = \(\frac{-1 - 2}{8 - 7} = \frac{-3}{1} = -3 \, \text{m/s}\).

5. **Fifth segment (from \( t = 8 \) to \( t = 10 \) seconds)**:
   - The line rises from -1 to 0.
   - Slope (velocity) = \(\frac{0 - (-1)}{10 - 8} = \frac{1}{2} = 0.5 \, \text{m/s}\).

### Part B: Create the \(a\)-\(t\) graph (acceleration vs. time graph)

Acceleration is the rate of change of velocity. We will calculate the acceleration for each time interval:

1. **From \( t = 0 \) to \( t = 3 \) seconds**:
   - Constant velocity, hence acceleration = 0.

2. **From \( t = 3 \) to \( t = 5 \) seconds**:
   - Change in velocity: \( -1 \, \text{m/s} - \frac{2}{3} \, \text{m/s} \).
   - Time interval: \( 2 \, \text{s} \).
   - Acceleration = \( \frac{-1 - \frac{2}{3}}{2} = \frac{-5/3}{2} = -\frac{5}{6} \, \text{m/s}^2 \).

3. **From \( t = 5 \) to \( t = 7 \) seconds**:
   - Change in velocity: \( 1 \, \text{m/s} - (-1) \, \text{m/s} = 2 \, \text{m/s} \).
   - Time interval: \( 2 \,

Learn how to solve the problem presented in the x-t graph by analyzing velocity and acceleration. This problem involves creating velocity-time and acceleration-time graphs based on position-time data. Suitable for high school students studying graphs and kinematics.

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