Distance and Displacement Practice Worksheet with Graph and Questions

Problem Overview:

The task involves calculating distance and displacement between various points on a grid. The grid is a coordinate system where each block represents 1 meter in length and height. The directions are defined as follows:
- \( +y \): North
- \( -y \): South
- \( +x \): East
- \( -x \): West

#### Key Concepts:
1. Distance: The total length of the path traveled, regardless of direction.
2. Displacement: The shortest distance from the starting point to the ending point, along with its direction.

We will solve each question step by step.

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Solution:

#### 1. Distance and Displacement from Point A to Point B
- Coordinates:
- \( A = (1, 0) \)
- \( B = (6, 0) \)
- Distance:
- Since both points lie on the same horizontal line (\( y = 0 \)), the distance is simply the difference in their \( x \)-coordinates:
\[
\text{Distance} = |6 - 1| = 5 \, \text{meters}
\]
- Displacement:
- Displacement is also the difference in \( x \)-coordinates, but it includes direction:
\[
\text{Displacement} = 5 \, \text{meters East}
\]

Answer:
\[
\boxed{5 \, \text{meters}, 5 \, \text{meters East}}
\]

---

#### 2. Distance and Displacement from Point A to Point E
- Coordinates:
- \( A = (1, 0) \)
- \( E = (4, 7) \)
- Distance:
- Use the Pythagorean theorem since the path is not horizontal or vertical:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - 1)^2 + (7 - 0)^2} = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( A \) to \( E \), which is \( \sqrt{58} \) meters. The direction can be described using the change in \( x \) and \( y \):
\[
\text{Displacement} = \sqrt{58} \, \text{meters, approximately } 7.62 \, \text{meters at an angle north-east}
\]

Answer:
\[
\boxed{\sqrt{58} \, \text{meters}, \sqrt{58} \, \text{meters North-East}}
\]

---

#### 3. Distance and Displacement from Point A to Point G
- Coordinates:
- \( A = (1, 0) \)
- \( G = (3, 0) \)
- Distance:
- Both points lie on the same horizontal line (\( y = 0 \)):
\[
\text{Distance} = |3 - 1| = 2 \, \text{meters}
\]
- Displacement:
- Displacement is the same as the distance, with direction:
\[
\text{Displacement} = 2 \, \text{meters East}
\]

Answer:
\[
\boxed{2 \, \text{meters}, 2 \, \text{meters East}}
\]

---

#### 4. Distance and Displacement from Point B to Point C
- Coordinates:
- \( B = (6, 0) \)
- \( C = (8, 3) \)
- Distance:
- Use the Pythagorean theorem:
\[
\text{Distance} = \sqrt{(8 - 6)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( B \) to \( C \):
\[
\text{Displacement} = \sqrt{13} \, \text{meters, approximately } 3.61 \, \text{meters at an angle north-east}
\]

Answer:
\[
\boxed{\sqrt{13} \, \text{meters}, \sqrt{13} \, \text{meters North-East}}
\]

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#### 5. Distance and Displacement if you travel through points \( A \to B \to D \)
- Path:
- \( A = (1, 0) \to B = (6, 0) \to D = (7, 5) \)
- Distance:
- From \( A \) to \( B \):
\[
\text{Distance}_{AB} = |6 - 1| = 5 \, \text{meters}
\]
- From \( B \) to \( D \):
\[
\text{Distance}_{BD} = \sqrt{(7 - 6)^2 + (5 - 0)^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \, \text{meters}
\]
- Total distance:
\[
\text{Total Distance} = 5 + \sqrt{26} \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( A \) to \( D \):
\[
\text{Displacement} = \sqrt{(7 - 1)^2 + (5 - 0)^2} = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61} \, \text{meters}
\]

Answer:
\[
\boxed{5 + \sqrt{26} \, \text{meters}, \sqrt{61} \, \text{meters North-East}}
\]

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#### 6. Distance and Displacement if you travel through points \( A \to G \to F \)
- Path:
- \( A = (1, 0) \to G = (3, 0) \to F = (3, 7) \)
- Distance:
- From \( A \) to \( G \):
\[
\text{Distance}_{AG} = |3 - 1| = 2 \, \text{meters}
\]
- From \( G \) to \( F \):
\[
\text{Distance}_{GF} = |7 - 0| = 7 \, \text{meters}
\]
- Total distance:
\[
\text{Total Distance} = 2 + 7 = 9 \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( A \) to \( F \):
\[
\text{Displacement} = \sqrt{(3 - 1)^2 + (7 - 0)^2} = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53} \, \text{meters}
\]

Answer:
\[
\boxed{9 \, \text{meters}, \sqrt{53} \, \text{meters North-East}}
\]

---

#### 7. Distance and Displacement if you travel through points \( H \to J \to I \to H \)
- Path:
- \( H = (3, -5) \to J = (8, -5) \to I = (4, -8) \to H = (3, -5) \)
- Distance:
- From \( H \) to \( J \):
\[
\text{Distance}_{HJ} = |8 - 3| = 5 \, \text{meters}
\]
- From \( J \) to \( I \):
\[
\text{Distance}_{JI} = \sqrt{(4 - 8)^2 + (-8 + 5)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{meters}
\]
- From \( I \) to \( H \):
\[
\text{Distance}_{IH} = \sqrt{(3 - 4)^2 + (-5 + 8)^2} = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \, \text{meters}
\]
- Total distance:
\[
\text{Total Distance} = 5 + 5 + \sqrt{10} = 10 + \sqrt{10} \, \text{meters}
\]
- Displacement:
- Since you return to the starting point \( H \), the displacement is:
\[
\text{Displacement} = 0 \, \text{meters}
\]

Answer:
\[
\boxed{10 + \sqrt{10} \, \text{meters}, 0 \, \text{meters}}
\]

---

#### 8. Distance and Displacement if you travel through points \( F \to E \to A \to G \)
- Path:
- \( F = (3, 7) \to E = (4, 7) \to A = (1, 0) \to G = (3, 0) \)
- Distance:
- From \( F \) to \( E \):
\[
\text{Distance}_{FE} = |4 - 3| = 1 \, \text{meter}
\]
- From \( E \) to \( A \):
\[
\text{Distance}_{EA} = \sqrt{(1 - 4)^2 + (0 - 7)^2} = \sqrt{(-3)^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \, \text{meters}
\]
- From \( A \) to \( G \):
\[
\text{Distance}_{AG} = |3 - 1| = 2 \, \text{meters}
\]
- Total distance:
\[
\text{Total Distance} = 1 + \sqrt{58} + 2 = 3 + \sqrt{58} \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( F \) to \( G \):
\[
\text{Displacement} = \sqrt{(3 - 3)^2 + (0 - 7)^2} = \sqrt{0 + 49} = 7 \, \text{meters South}
\]

Answer:
\[
\boxed{3 + \sqrt{58} \, \text{meters}, 7 \, \text{meters South}}
\]

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#### 9. Distance and Displacement if you travel through points \( A \to B \to C \to D \to E \)
- Path:
- \( A = (1, 0) \to B = (6, 0) \to C = (8, 3) \to D = (7, 5) \to E = (4, 7) \)
- Distance:
- From \( A \) to \( B \):
\[
\text{Distance}_{AB} = |6 - 1| = 5 \, \text{meters}
\]
- From \( B \) to \( C \):
\[
\text{Distance}_{BC} = \sqrt{(8 - 6)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \, \text{meters}
\]
- From \( C \) to \( D \):
\[
\text{Distance}_{CD} = \sqrt{(7 - 8)^2 + (5 - 3)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \, \text{meters}
\]
- From \( D \) to \( E \):
\[
\text{Distance}_{DE} = \sqrt{(4 - 7)^2 + (7 - 5)^2} = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \, \text{meters}
\]
- Total distance:
\[
\text{Total Distance} = 5 + \sqrt{13} + \sqrt{5} + \sqrt{13} = 5 + 2\sqrt{13} + \sqrt{5} \, \text{meters}
\]
- Displacement:
- Displacement is the straight-line distance from \( A \) to \( E \):
\[
\text{Displacement} = \sqrt{(4 - 1)^2 + (7 - 0)^2} = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \, \text{meters}
\]

Answer:
\[
\boxed{5 + 2\sqrt{13} + \sqrt{5} \, \text{meters}, \sqrt{58} \, \text{meters North-East}}
\]

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#### 10. Choose any 4 points on the graph to travel through and find the distance and displacement
- Points Chosen: \( H \to J \to I \to H \)
- Solution:
- This was already solved in Question 7.
- Distance: \( 10 + \sqrt{10} \, \text{meters} \)
- Displacement: \( 0 \, \text{meters} \)

Answer:
\[
\boxed{10 + \sqrt{10} \, \text{meters}, 0 \, \text{meters}}
\]

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Final Answers:

1. \( \boxed{5 \, \text{meters}, 5 \, \text{meters East}} \)
2. \( \boxed{\sqrt{58} \, \text{meters}, \sqrt{58} \, \text{meters North-East}} \)
3. \( \boxed{2 \, \text{meters}, 2 \, \text{meters East}} \)
4. \( \boxed{\sqrt{13} \, \text{meters}, \sqrt{13} \, \text{meters North-East}} \)
5. \( \boxed{5 + \sqrt{26} \, \text{meters}, \sqrt{61} \, \text{meters North-East}} \)
6. \( \boxed{9 \, \text{meters}, \sqrt{53} \, \text{meters North-East}} \)
7. \( \boxed{10 + \sqrt{10} \, \text{meters}, 0 \, \text{meters}} \)
8. \( \boxed{3 + \sqrt{58} \, \text{meters}, 7 \, \text{meters South}} \)
9. \( \boxed{5 + 2\sqrt{13} + \sqrt{5} \, \text{meters}, \sqrt{58} \, \text{meters North-East}} \)
10. \( \boxed{10 + \sqrt{10} \, \text{meters}, 0 \, \text{meters}} \)