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Let's solve the distance and displacement for each problem listed in the image:

### 1. Amy runs 2 miles south, then turns around and runs 3 miles north.
- **Distance**: Total distance covered is the sum of both legs of the run:
  \[
  2 \, \text{miles (south)} + 3 \, \text{miles (north)} = 5 \, \text{miles}.
  \]
- **Displacement**: Displacement is the straight-line distance from the starting point to the ending point, considering direction. Since Amy runs 2 miles south, then 3 miles north, her final displacement is:
  \[
  3 \, \text{miles} - 2 \, \text{miles} = 1 \, \text{mile north}.
  \]

### 2. Jermaine runs exactly 2 laps around a 400-meter track.
- **Distance**: The distance for one lap is 400 meters. Therefore, for two laps:
  \[
  2 \times 400 \, \text{meters} = 800 \, \text{meters}.
  \]
- **Displacement**: Since Jermaine ends up where he started after 2 laps, his displacement is zero.

### 3. Derrick crawls 4 feet south, then turns east and crawls 6 feet.
- **Distance**: The total distance crawled is the sum of both parts:
  \[
  4 \, \text{feet (south)} + 6 \, \text{feet (east)} = 10 \, \text{feet}.
  \]
- **Displacement**: Displacement is the straight-line distance between the start and the end point. This forms a right triangle where:
  - One leg = 4 feet (south),
  - Other leg = 6 feet (east),
  - Use the Pythagorean theorem to find the displacement:
  \[
  \text{Displacement} = \sqrt{(4^2 + 6^2)} = \sqrt{(16 + 36)} = \sqrt{52} \approx 7.21 \, \text{feet}.
  \]

### 4. Ray runs 30 feet north, 30 feet west, and 30 feet south.
- **Distance**: The total distance is:
  \[
  30 \, \text{feet (north)} + 30 \, \text{feet (west)} + 30 \, \text{feet (south)} = 90 \, \text{feet}.
  \]
- **Displacement**: Since Ray runs north and then back south the same amount, the vertical displacement is zero. His only displacement is 30 feet west.

### 5. Alex walks east for 3 km, stops for a break, and then runs in the same direction for 4 km.
- **Distance**: The total distance covered is:
  \[
  3 \, \text{km (walking)} + 4 \, \text{km (running)} = 7 \, \text{km}.
  \]
- **Displacement**: Since Alex is walking and running in the same direction (east), his displacement is also:
  \[
  7 \, \text{km east}.
  \]

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Would you like further details or have any questions on these solutions?

Here are 5 related questions:
1. How does displacement differ from distance in circular motion?
2. If Amy had run 2 miles west instead of south, how would that affect her displacement?
3. What is the significance of zero displacement in Jermaine's case?
4. How would Ray’s displacement change if he ran 30 feet east after his final south run?
5. Can displacement ever be greater than distance?

**Tip**: Displacement considers both magnitude and direction, while distance is the total path length traveled, regardless of direction.

Determine the distance and displacement for Amy, Jermaine, Derrick, Ray, and Alex based on their movements.

Learn how to calculate both the distance and displacement for various paths, including examples of linear and circular motions. This tutorial will walk through the movements of Amy, Jermaine, Derrick, Ray, and Alex with step-by-step solutions, including using the Pythagorean Theorem for complex paths.

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