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### 2.1 Define a vector
A vector is a mathematical object that has both magnitude and direction. Vectors are often represented as arrows in a coordinate system, where the length of the arrow represents the magnitude and the arrowhead indicates the direction. Common examples of vectors include displacement, velocity, and force.
### 2.2 Training Session Analysis
#### 2.2.1 Draw a vector scale diagram
Considering the scale of [tex]\( 1 \, \text{cm} = 20 \, \text{m} \)[/tex], we can represent the displacements as follows:
- [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: 120 m East → [tex]\( \frac{120 \, \text{m}}{20 \, \text{m/cm}} = 6 \, \text{cm} \)[/tex] East
- [tex]\( B \)[/tex] to [tex]\( C \)[/tex]: 80 m West → [tex]\( \frac{80 \, \text{m}}{20 \, \text{m/cm}} = 4 \, \text{cm} \)[/tex] West
- [tex]\( C \)[/tex] to [tex]\( D \)[/tex]: 60 m West → [tex]\( \frac{60 \, \text{m}}{20 \, \text{m/cm}} = 3 \, \text{cm} \)[/tex] West
To create the diagram:
1. Draw an arrow 6 cm to the right (East) from point [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
2. From point [tex]\( B \)[/tex], draw an arrow 4 cm to the left (West) to point [tex]\( C \)[/tex].
3. From point [tex]\( C \)[/tex], draw another arrow 3 cm to the left (West) to point [tex]\( D \)[/tex].
#### 2.2.2 Draw and indicate the resultant vector
The resultant vector is the net change in position from point [tex]\( A \)[/tex] to point [tex]\( D \)[/tex]:
- Start at point [tex]\( A \)[/tex].
- The resultant vector extends from [tex]\( A \)[/tex] to [tex]\( D \)[/tex], combining all intermediate movements.
- Given the previous vectors, it should be drawn directly from [tex]\( A \)[/tex] to [tex]\( D \)[/tex], indicating the final position relative to the start.
#### 2.2.3 Calculate the total distance covered by the athlete
The total distance covered is the sum of the absolute values of all the segments run, regardless of direction:
[tex]\[ \text{Total Distance} = 120 \, \text{m} + 80 \, \text{m} + 60 \, \text{m} = 260 \, \text{m} \][/tex]
#### 2.2.4 What is her change in position at [tex]\( D \)[/tex], relative to [tex]\( A \)[/tex]?
The net displacement is the sum of the directional movements:
1. [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: 120 m East
2. [tex]\( B \)[/tex] to [tex]\( C \)[/tex]: 80 m West
3. [tex]\( C \)[/tex] to [tex]\( D \)[/tex]: 60 m West
The net displacement:
[tex]\[ 120 \, \text{m} - 80 \, \text{m} - 60 \, \text{m} = 120 \, \text{m} - 140 \, \text{m} = -20 \, \text{m} \][/tex]
This means the athlete ends up 20 m West of her starting position.
#### 2.2.5 Calculate the average velocity of the athlete
Average velocity is defined as the total displacement divided by the total time taken.
1. The total displacement is -20 m (20 m West).
2. The total time taken is 2 minutes, which needs to be converted to seconds for velocity calculation:
[tex]\[ 2 \, \text{minutes} \times 60 \, \text{seconds/minute} = 120 \, \text{seconds} \][/tex]
Average velocity is then calculated as:
[tex]\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{-20 \, \text{m}}{120 \, \text{s}} = -\frac{1}{6} \, \text{m/s} \, \approx \, -0.167 \, \text{m/s} \][/tex]
The negative sign indicates the direction is towards the West.
### Summary of Answers
1. Definition of a vector (2 marks)
2. Scale diagram and resultant vector (4 marks)
3. Total distance covered: 260 m (1 mark)
4. Change in position: 20 m West (1 mark)
5. Average velocity: -0.167 m/s (4 marks)
[Total: 12 marks]
A vector is a mathematical object that has both magnitude and direction. Vectors are often represented as arrows in a coordinate system, where the length of the arrow represents the magnitude and the arrowhead indicates the direction. Common examples of vectors include displacement, velocity, and force.
### 2.2 Training Session Analysis
#### 2.2.1 Draw a vector scale diagram
Considering the scale of [tex]\( 1 \, \text{cm} = 20 \, \text{m} \)[/tex], we can represent the displacements as follows:
- [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: 120 m East → [tex]\( \frac{120 \, \text{m}}{20 \, \text{m/cm}} = 6 \, \text{cm} \)[/tex] East
- [tex]\( B \)[/tex] to [tex]\( C \)[/tex]: 80 m West → [tex]\( \frac{80 \, \text{m}}{20 \, \text{m/cm}} = 4 \, \text{cm} \)[/tex] West
- [tex]\( C \)[/tex] to [tex]\( D \)[/tex]: 60 m West → [tex]\( \frac{60 \, \text{m}}{20 \, \text{m/cm}} = 3 \, \text{cm} \)[/tex] West
To create the diagram:
1. Draw an arrow 6 cm to the right (East) from point [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
2. From point [tex]\( B \)[/tex], draw an arrow 4 cm to the left (West) to point [tex]\( C \)[/tex].
3. From point [tex]\( C \)[/tex], draw another arrow 3 cm to the left (West) to point [tex]\( D \)[/tex].
#### 2.2.2 Draw and indicate the resultant vector
The resultant vector is the net change in position from point [tex]\( A \)[/tex] to point [tex]\( D \)[/tex]:
- Start at point [tex]\( A \)[/tex].
- The resultant vector extends from [tex]\( A \)[/tex] to [tex]\( D \)[/tex], combining all intermediate movements.
- Given the previous vectors, it should be drawn directly from [tex]\( A \)[/tex] to [tex]\( D \)[/tex], indicating the final position relative to the start.
#### 2.2.3 Calculate the total distance covered by the athlete
The total distance covered is the sum of the absolute values of all the segments run, regardless of direction:
[tex]\[ \text{Total Distance} = 120 \, \text{m} + 80 \, \text{m} + 60 \, \text{m} = 260 \, \text{m} \][/tex]
#### 2.2.4 What is her change in position at [tex]\( D \)[/tex], relative to [tex]\( A \)[/tex]?
The net displacement is the sum of the directional movements:
1. [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: 120 m East
2. [tex]\( B \)[/tex] to [tex]\( C \)[/tex]: 80 m West
3. [tex]\( C \)[/tex] to [tex]\( D \)[/tex]: 60 m West
The net displacement:
[tex]\[ 120 \, \text{m} - 80 \, \text{m} - 60 \, \text{m} = 120 \, \text{m} - 140 \, \text{m} = -20 \, \text{m} \][/tex]
This means the athlete ends up 20 m West of her starting position.
#### 2.2.5 Calculate the average velocity of the athlete
Average velocity is defined as the total displacement divided by the total time taken.
1. The total displacement is -20 m (20 m West).
2. The total time taken is 2 minutes, which needs to be converted to seconds for velocity calculation:
[tex]\[ 2 \, \text{minutes} \times 60 \, \text{seconds/minute} = 120 \, \text{seconds} \][/tex]
Average velocity is then calculated as:
[tex]\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{-20 \, \text{m}}{120 \, \text{s}} = -\frac{1}{6} \, \text{m/s} \, \approx \, -0.167 \, \text{m/s} \][/tex]
The negative sign indicates the direction is towards the West.
### Summary of Answers
1. Definition of a vector (2 marks)
2. Scale diagram and resultant vector (4 marks)
3. Total distance covered: 260 m (1 mark)
4. Change in position: 20 m West (1 mark)
5. Average velocity: -0.167 m/s (4 marks)
[Total: 12 marks]
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