https://file.aiquickdraw.com/mathbot/file/172596868486441bqzyw6.jpg

### Part I: Fill in the Table

Let's fill in the table based on the concepts of motion graphs.

#### 1. **Position vs. Time Graph**

- **Positive Slope:**
  - **Shape:** A straight line slanting upwards from left to right.
  - **Motion Interpretation:** The object is moving forward with constant velocity.

- **Negative Slope:**
  - **Shape:** A straight line slanting downwards from left to right.
  - **Motion Interpretation:** The object is moving backward (or in the opposite direction) with constant velocity.

- **Zero Slope:**
  - **Shape:** A horizontal straight line.
  - **Motion Interpretation:** The object is at rest (no change in position).

#### 2. **Velocity vs. Time Graph**

- **Positive Slope:**
  - **Shape:** A straight line slanting upwards from left to right.
  - **Motion Interpretation:** The object is accelerating (increasing velocity).

- **Negative Slope:**
  - **Shape:** A straight line slanting downwards from left to right.
  - **Motion Interpretation:** The object is decelerating (decreasing velocity).

- **Zero Slope:**
  - **Shape:** A horizontal straight line.
  - **Motion Interpretation:** The object is moving with constant velocity (no change in velocity).

### Part II: Problem-Solving

Let's solve the given problems one by one.

#### 1. **Sara's Acceleration Problem**

Sara starts from rest (initial velocity \( u = 0 \)) and accelerates at a rate of \( a = 2 \, \text{m/s}^2 \) for \( t = 10 \, \text{seconds} \).

- **Final Velocity \( v \):**
  \[
  v = u + at = 0 + (2 \times 10) = 20 \, \text{m/s}
  \]

- **Distance Traveled \( s \):**
  \[
  s = ut + \frac{1}{2}at^2 = 0 \times 10 + \frac{1}{2} \times 2 \times 10^2 = \frac{1}{2} \times 2 \times 100 = 100 \, \text{meters}
  \]

Sara travels 100 meters.

#### 2. **Alice's Constant Velocity Problem**

Alice maintains a constant velocity of \( v = 15 \, \text{m/s} \) for \( t = 20 \, \text{seconds} \).

- **Distance Traveled \( s \):**
  \[
  s = vt = 15 \times 20 = 300 \, \text{meters}
  \]

Alice travels 300 meters.

#### 3. **Emily's Displacement Problem**

Emily stands still for 5 minutes (which is \( t = 5 \times 60 = 300 \, \text{seconds} \)).

- **Displacement:** Since she is standing still, her displacement is \( 0 \, \text{meters} \).

### Do you need further details or have any questions? Here are some related questions to expand on this:

1. How would the position vs. time graph look for an object that starts at a non-zero position and moves with constant velocity?
2. What happens to the velocity vs. time graph if an object accelerates uniformly, then suddenly stops?
3. How can the area under a velocity vs. time graph help determine the distance traveled?
4. What type of motion would be represented by a position vs. time graph that curves upwards?
5. How does a negative slope in a velocity vs. time graph relate to real-life situations?

**Tip:** Always pay attention to the initial conditions given in a problem (like starting from rest) as they are crucial in determining how you approach the calculations.

Interpret motion graphs and solve motion-related problems, including analyzing slopes in Position vs. Time and Velocity vs. Time graphs.

Learn how to interpret motion graphs such as Position vs. Time and Velocity vs. Time. Solve real-world motion problems involving acceleration, constant velocity, and rest. This lesson includes essential kinematic formulas like v = u + at and s = ut + (1/2)at^2. Suitable for Grades 9-12.

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