Distance and Displacement Worksheet with six word problems requiring students to calculate total distance traveled and net displacement using directional movements.

Let's solve each problem step by step, explaining the difference between distance and displacement:

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🔹 Key Concepts:

- Distance: The total path traveled (scalar quantity – only magnitude).
- Displacement: The straight-line distance from start to end point, including direction (vector quantity – magnitude and direction).

We'll use a coordinate system:
- North = +y
- South = -y
- East = +x
- West = -x

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✔ Problem 1: Chance

> Chance drives her scooter 7 km north. She stops for lunch and then drives 5 km east.

#### Distance:
Total path covered = 7 km + 5 km = 12 km

#### Displacement:
She starts at origin (0,0), goes 7 km north → (0,7), then 5 km east → (5,7)

Use Pythagoras’ theorem:
$$
\text{Displacement} = \sqrt{(5)^2 + (7)^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.6 \text{ km}
$$
Direction: Northeast (specifically, tan⁻¹(7/5) ≈ 54.5° north of east)

✔ Answer:
- Distance: 12 km
- Displacement: ~8.6 km, 54.5° north of east

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✔ Problem 2: Anthony

> Walks 1 km east, then 1 km south, then 1 km east again.

#### Distance:
1 + 1 + 1 = 3 km

#### Displacement:
Start at (0,0)
- 1 km east → (1,0)
- 1 km south → (1,-1)
- 1 km east → (2,-1)

So final position is (2,-1)

Displacement = √(2² + (-1)²) = √(4 + 1) = √5 ≈ 2.24 km

Direction: tan⁻¹(1/2) ≈ 26.6° south of east

✔ Answer:
- Distance: 3 km
- Displacement: ~2.24 km, 26.6° south of east

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✔ Problem 3: Justin (Fishing Trip)

> 12 km south, 4 km west, 1 km north

#### Distance:
12 + 4 + 1 = 17 km

#### Displacement:
Start at (0,0)
- 12 km south → (0, -12)
- 4 km west → (-4, -12)
- 1 km north → (-4, -11)

Final position: (-4, -11)

Displacement = √[(-4)² + (-11)²] = √(16 + 121) = √137 ≈ 11.7 km

Direction: tan⁻¹(11/4) ≈ 70.0° south of west

✔ Answer:
- Distance: 17 km
- Displacement: ~11.7 km, 70° south of west

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✔ Problem 4: Tara (Camel Safari)

> 5 km north, 3 km east, 1 km north

#### Distance:
5 + 3 + 1 = 9 km

#### Displacement:
Start at (0,0)
- 5 km north → (0,5)
- 3 km east → (3,5)
- 1 km north → (3,6)

Final position: (3,6)

Displacement = √(3² + 6²) = √(9 + 36) = √45 ≈ 6.71 km

Direction: tan⁻¹(6/3) = tan⁻¹(2) ≈ 63.4° north of east

✔ Answer:
- Distance: 9 km
- Displacement: ~6.71 km, 63.4° north of east

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✔ Problem 5: Alex (Dirt Bike)

> 700 m north, 300 m east, 400 m north, 600 m west, 1200 m south, 100 m north

All in meters. Let’s compute net movement.

#### Net North-South:
- 700 N + 400 N = 1100 N
- Then 1200 S → so 1100 N - 1200 S = -100 m (i.e., 100 m south)
- Then +100 m north → 0 m (net)

→ Net Y = 0 m

#### Net East-West:
- 300 E
- 600 W → 300 E - 600 W = -300 m (i.e., 300 m west)

→ Net X = -300 m

So final position: (-300, 0)

#### Distance:
Add all segments:
700 + 300 + 400 + 600 + 1200 + 100 = 3300 m

#### Displacement:
From (0,0) to (-300, 0) → 300 m west

✔ Answer:
- Distance: 3300 m
- Displacement: 300 m west

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✔ Problem 6: Jose (Moped)

> Travels 3 km south, then 4 km east

#### Distance:
3 + 4 = 7 km

#### Displacement:
Start at (0,0)
- 3 km south → (0, -3)
- 4 km east → (4, -3)

Displacement = √(4² + (-3)²) = √(16 + 9) = √25 = 5 km

Direction: tan⁻¹(3/4) = 36.9° south of east

But question asks: How far does he need to go to get back to where he started?

That’s the magnitude of displacement: 5 km, directly opposite (northwest direction).

✔ Answer:
- He needs to travel 5 km in the direction 36.9° north of west (or equivalently, northwest) to return.

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✔ Final Answers Summary:

| Question | Distance | Displacement |
|--------|---------|-------------|
| 1 | 12 km | ~8.6 km, 54.5° N of E |
| 2 | 3 km | ~2.24 km, 26.6° S of E |
| 3 | 17 km | ~11.7 km, 70° S of W |
| 4 | 9 km | ~6.71 km, 63.4° N of E |
| 5 | 3300 m | 300 m west |
| 6 | 7 km | 5 km (must travel 5 km back to start) |

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📌 Notes on Grid Plotting (as requested):

For each problem, plot on grid paper:
- Use axes with N/S/E/W directions.
- Scale: e.g., 1 cm = 1 km or 1 cm = 100 m depending on problem.
- Draw arrows for each leg of motion.
- Draw a straight line from start to end for displacement.

Example for Q1:
- Start at origin
- Arrow up 7 cm (N)
- Arrow right 5 cm (E)
- Draw diagonal from start to end point → this is displacement.

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Let me know if you'd like a visual sketch or printable version!