Worksheet with five problems on calculating distance and displacement, including examples like walking, driving, and running.

Let's go through each problem on the Distance and Displacement Worksheet and evaluate both the distance (total path traveled) and displacement (straight-line change in position from start to end point), with explanations.

---

🔹 Key Definitions:

- Distance: The total length of the path traveled. It is a scalar quantity (only magnitude).
- Displacement: The straight-line distance from the starting point to the ending point, including direction. It is a vector quantity (magnitude and direction).

---

✔ Problem 1: Stan walks 10 km west to the grocery store, then 10 km east back home.

- Distance:
He walks 10 km + 10 km = 20 km
✔️ Correct

- Displacement:
He starts at home → goes 10 km west → returns 10 km east → ends at home.
So net movement = 0 km
Displacement = 0 km (no direction needed since it's zero)
✔️ Correct

> ✔ Answer: Distance = 20 km, Displacement = 0 km

---

✔ Problem 2: Tyler drives 50 km north, then 30 km south.

- Distance:
Total path = 50 km + 30 km = 80 km
✔️ Correct

- Displacement:
North is positive, South is negative.
Net movement = 50 km (N) – 30 km (S) = 20 km north
✘ Incorrect: The answer says "20km south" — this is wrong.

> ✘ Correction: Displacement is 20 km north, not south.

> ✔ Correct Answer: Distance = 80 km, Displacement = 20 km north

---

✔ Problem 3: Joey drives 7 km north, then 5 km east.

- Distance:
Total path = 7 km + 5 km = 12 km
✔️ Correct

- Displacement:
This is a right triangle: 7 km north and 5 km east.
Use Pythagoras' theorem:
$$
\text{Displacement} = \sqrt{(7)^2 + (5)^2} = \sqrt{49 + 25} = \sqrt{74} \approx 8.6 \text{ km}
$$
Direction: northeast (specifically, angle from north or east can be found using tan⁻¹(5/7)).

But the worksheet says: "7 - 5 = 2 km East" — this is wrong because:
- You can't subtract perpendicular vectors like that.
- North and East are perpendicular directions.

> ✘ Incorrect: The displacement is not 2 km east.

> ✔ Correct Answer:
> Distance = 12 km
> Displacement ≈ 8.6 km in a northeast direction (or more precisely, about 35.5° east of north)

---

✔ Problem 4: Brittney runs 400 m around a circular track and returns to start.

- Distance:
She ran one full lap = 400 m
✔️ Correct

- Displacement:
She started and ended at the same point → no net change in position
So displacement = 0 m
✔️ Correct

> ✔ Answer: Distance = 400 m, Displacement = 0 m

---

✔ Problem 5: Anthony walks 4 km east, then 1 km west.

- Distance:
Total path = 4 km + 1 km = 5 km
✔️ Correct

- Displacement:
Net movement = 4 km east – 1 km west = 3 km east
The answer says "3km EW" — this notation is unclear.

“EW” likely means “east-west”, but it’s ambiguous.
Better to say: 3 km east

> ✘ Improvement Needed: Instead of "3km EW", write 3 km east

> ✔ Correct Answer: Distance = 5 km, Displacement = 3 km east

---

📌 Summary of Corrections:

| Problem | Distance | Displacement (Corrected) | Error |
|--------|----------|---------------------------|-------|
| 1 | 20 km | 0 km | ✔ Correct |
| 2 | 80 km | 20 km north | ✘ Said "south" |
| 3 | 12 km | ≈8.6 km NE | ✘ Used subtraction instead of vector addition |
| 4 | 400 m | 0 m | ✔ Correct |
| 5 | 5 km | 3 km east | ✘ "EW" is vague |

---

✔ Final Answers (Corrected):

1. Distance: 20 km
Displacement: 0 km

2. Distance: 80 km
Displacement: 20 km north

3. Distance: 12 km
Displacement: √(7² + 5²) = √74 ≈ 8.6 km, direction: ~35.5° east of north

4. Distance: 400 m
Displacement: 0 m

5. Distance: 5 km
Displacement: 3 km east

---

Let me know if you'd like a diagram or explanation for any specific problem!