https://r2.erweima.ai/mathbot/image/1726541971749q1ew8kq2.jpg

Let's work through each problem related to distance and displacement from the image:

### 1. **Amy runs 2 miles south, then turns around and runs 3 miles north.**

- **Distance:** Distance is the total path covered, so Amy ran 2 miles south and 3 miles north. Total distance = 2 + 3 = **5 miles**.
- **Displacement:** Displacement is the straight-line distance between the starting and ending points, with direction. She ends 1 mile north of her starting point. Displacement = **1 mile north**.

### 2. **Jermaine runs exactly 2 laps around a 400-meter track.**

- **Distance:** One lap around the track is 400 meters. Two laps = 2 × 400 = **800 meters**.
- **Displacement:** Since Jermaine ends up at the same point where he started, the displacement = **0 meters**.

### 3. **Derrick crawls 4 feet south, and then turns east and crawls 6 feet.**

- **Distance:** The total distance Derrick crawls is 4 + 6 = **10 feet**.
- **Displacement:** We can use the Pythagorean theorem to find the displacement. His movement forms a right triangle, where the legs are 4 feet and 6 feet.
  \[
  \text{Displacement} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21 \text{ feet}.
  \]
  The displacement direction is south-east, but the exact angle can be calculated using trigonometry.

### 4. **Ray runs 30 feet north, 30 feet west, and then 30 feet south.**

- **Distance:** The total distance is 30 + 30 + 30 = **90 feet**.
- **Displacement:** Ray's final displacement can be visualized as running 30 feet north and 30 feet south, canceling out the north-south movement. So, the only net movement is 30 feet west. Displacement = **30 feet west**.

### 5. **Alex walks east for 3 km, stops for a break, and then runs in the same direction for 4 km before he stops.**

- **Distance:** Total distance = 3 + 4 = **7 km**.
- **Displacement:** Since Alex moves in a straight line in the same direction, his displacement is also **7 km east**.

Would you like me to go over any specific details or vector diagrams?

Here are 5 related questions to further practice:

1. What is the difference between distance and displacement?
2. How does displacement differ when an object moves in a straight line versus along a curved path?
3. If someone runs in a circle and ends up where they started, how would you explain their displacement?
4. How does the Pythagorean theorem help calculate displacement when movement occurs in two dimensions?
5. What would happen to the displacement if Amy, in problem 1, ran 3 miles south instead of north?

**Tip:** Remember that distance is a scalar quantity (it only has magnitude), whereas displacement is a vector (it has both magnitude and direction).

Determine the distance and displacement of each of the following problems. 1. Amy runs 2 miles south, then turns around and runs 3 miles north. 2. Jermaine runs exactly 2 laps around a 400-meter track. 3. Derrick crawls 4 feet south, and then turns east and crawls 6 feet. 4. Ray runs 30 feet north, 30 feet west, and then 30 feet south. 5. Alex walks east for 3 km, stops for a break, and then runs in the same direction for 4 km before he stops.

Learn how to calculate distance and displacement through real-world examples involving Amy, Jermaine, Derrick, Ray, and Alex. This detailed problem set covers vector analysis and uses the Pythagorean theorem to calculate movement in two dimensions.

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