Answer :
Let's address each of the parts of the question in a clear, step-by-step manner:
### 2.1. Define the Term Vector
A vector is a quantity that has both magnitude and direction. Common examples include force, velocity, and displacement. Unlike scalar quantities, which have only magnitude (e.g., mass or temperature), vectors convey more information by also indicating direction.
### 2.2. Calculate the Resultant Force Acting on the Motorbike
The driving force produced by the motorbike is 500 N (Newtons), and when the brakes are applied, a frictional (braking) force of 150 N acts in the opposite direction. To find the resultant force, we subtract the braking force from the driving force:
[tex]\[ \text{Resultant Force} = \text{Driving Force} - \text{Braking Force} \][/tex]
[tex]\[ \text{Resultant Force} = 500 \, \text{N} - 150 \, \text{N} = 350 \, \text{N} \][/tex]
### 2.3. Write Down the Total Displacement for the Entire Journey
The bike travels 160 km westwards against the wind and then 160 km westwards with the wind. Therefore, the total displacement is the sum of these two distances:
[tex]\[ \text{Total Displacement} = 160 \, \text{km} + 160 \, \text{km} = 320 \, \text{km} \][/tex]
### 2.4. Calculate the Average Speed of the Motorbike for the Entire Journey
To find the average speed, we need to consider both the total displacement and the total time taken. The time taken for each part of the journey is given:
- 2 hours against the wind
- 1.67 hours with the wind
The total time for the journey is:
[tex]\[ \text{Total Time} = 2 \, \text{hours} + 1.67 \, \text{hours} = 3.67 \, \text{hours} \][/tex]
Average speed is calculated as the total displacement divided by the total time:
[tex]\[ \text{Average Speed} = \frac{\text{Total Displacement}}{\text{Total Time}} \][/tex]
[tex]\[ \text{Average Speed} = \frac{320 \, \text{km}}{3.67 \, \text{hours}} \approx 87.19 \, \text{km/h} \][/tex]
### 2.5. Calculate the Magnitude of the Actual Velocity of the Motorbike
First, we need to determine the actual velocities of the motorbike, both against and with the wind. The speed of the wind is given as [tex]\( 8 \, \text{km/h} \)[/tex].
#### Against the Wind
Without wind, the velocity of the motorbike against the wind is:
[tex]\[ \text{Actual Velocity Against Wind} = \left(\frac{\text{distance}}{\text{time}}\right) - \text{wind speed} \][/tex]
[tex]\[ \text{Actual Velocity Against Wind} = \left(\frac{160 \, \text{km}}{2 \, \text{hours}}\right) - 8 \, \text{km/h} \][/tex]
[tex]\[ \text{Actual Velocity Against Wind} = 80 \, \text{km/h} - 8 \, \text{km/h} = 72 \, \text{km/h} \][/tex]
#### With the Wind
Without wind, the velocity of the motorbike with the wind is:
[tex]\[ \text{Actual Velocity With Wind} = \left(\frac{\text{distance}}{\text{time}}\right) + \text{wind speed} \][/tex]
[tex]\[ \text{Actual Velocity With Wind} = \left(\frac{160 \, \text{km}}{1.67 \, \text{hours}}\right) + 8 \, \text{km/h} \][/tex]
[tex]\[ \text{Actual Velocity With Wind} \approx 95.81 \, \text{km/h} + 8 \, \text{km/h} \approx 103.81 \, \text{km/h} \][/tex]
Hence, the actual velocities of the motorbike are approximately:
- 72.0 km/h against the wind
- 103.81 km/h with the wind
### 2.1. Define the Term Vector
A vector is a quantity that has both magnitude and direction. Common examples include force, velocity, and displacement. Unlike scalar quantities, which have only magnitude (e.g., mass or temperature), vectors convey more information by also indicating direction.
### 2.2. Calculate the Resultant Force Acting on the Motorbike
The driving force produced by the motorbike is 500 N (Newtons), and when the brakes are applied, a frictional (braking) force of 150 N acts in the opposite direction. To find the resultant force, we subtract the braking force from the driving force:
[tex]\[ \text{Resultant Force} = \text{Driving Force} - \text{Braking Force} \][/tex]
[tex]\[ \text{Resultant Force} = 500 \, \text{N} - 150 \, \text{N} = 350 \, \text{N} \][/tex]
### 2.3. Write Down the Total Displacement for the Entire Journey
The bike travels 160 km westwards against the wind and then 160 km westwards with the wind. Therefore, the total displacement is the sum of these two distances:
[tex]\[ \text{Total Displacement} = 160 \, \text{km} + 160 \, \text{km} = 320 \, \text{km} \][/tex]
### 2.4. Calculate the Average Speed of the Motorbike for the Entire Journey
To find the average speed, we need to consider both the total displacement and the total time taken. The time taken for each part of the journey is given:
- 2 hours against the wind
- 1.67 hours with the wind
The total time for the journey is:
[tex]\[ \text{Total Time} = 2 \, \text{hours} + 1.67 \, \text{hours} = 3.67 \, \text{hours} \][/tex]
Average speed is calculated as the total displacement divided by the total time:
[tex]\[ \text{Average Speed} = \frac{\text{Total Displacement}}{\text{Total Time}} \][/tex]
[tex]\[ \text{Average Speed} = \frac{320 \, \text{km}}{3.67 \, \text{hours}} \approx 87.19 \, \text{km/h} \][/tex]
### 2.5. Calculate the Magnitude of the Actual Velocity of the Motorbike
First, we need to determine the actual velocities of the motorbike, both against and with the wind. The speed of the wind is given as [tex]\( 8 \, \text{km/h} \)[/tex].
#### Against the Wind
Without wind, the velocity of the motorbike against the wind is:
[tex]\[ \text{Actual Velocity Against Wind} = \left(\frac{\text{distance}}{\text{time}}\right) - \text{wind speed} \][/tex]
[tex]\[ \text{Actual Velocity Against Wind} = \left(\frac{160 \, \text{km}}{2 \, \text{hours}}\right) - 8 \, \text{km/h} \][/tex]
[tex]\[ \text{Actual Velocity Against Wind} = 80 \, \text{km/h} - 8 \, \text{km/h} = 72 \, \text{km/h} \][/tex]
#### With the Wind
Without wind, the velocity of the motorbike with the wind is:
[tex]\[ \text{Actual Velocity With Wind} = \left(\frac{\text{distance}}{\text{time}}\right) + \text{wind speed} \][/tex]
[tex]\[ \text{Actual Velocity With Wind} = \left(\frac{160 \, \text{km}}{1.67 \, \text{hours}}\right) + 8 \, \text{km/h} \][/tex]
[tex]\[ \text{Actual Velocity With Wind} \approx 95.81 \, \text{km/h} + 8 \, \text{km/h} \approx 103.81 \, \text{km/h} \][/tex]
Hence, the actual velocities of the motorbike are approximately:
- 72.0 km/h against the wind
- 103.81 km/h with the wind
