Answer :
Sure, let's break down this problem step by step.
### Question (ii): Show that the time taken by the missile to reach the aeroplane is...
1. Given Data:
- Angle of missile launch, [tex]\(\theta = 35^\circ\)[/tex]
- Acceleration of missile, [tex]\(a_{\text{missile}} = 50 \, \text{m/s}^2\)[/tex]
- Velocity of the aeroplane, [tex]\(v_{\text{aeroplane}} = 150 \, \text{m/s}\)[/tex]
2. Convert Angle to Radians:
The angle in radians is given as:
[tex]\[ \theta_{\text{rad}} = 0.6108652381980153 \, \text{radians} \][/tex]
3. Horizontal Component of Missile’s Acceleration:
The horizontal component of the missile's acceleration is calculated using the formula:
[tex]\[ a_{\text{horizontal}} = a_{\text{missile}} \cdot \cos(\theta) \][/tex]
Substituting the values:
[tex]\[ a_{\text{horizontal}} = 50 \times \cos(0.6108652381980153) = 40.95760221444959 \, \text{m/s}^2 \][/tex]
4. Horizontal Velocity of the Missile:
To reach the aeroplane, the horizontal component of the missile's velocity must equal the aeroplane’s horizontal velocity:
[tex]\[ v_{\text{horizontal}\_\text{missile}} = 150 \, \text{m/s} \][/tex]
5. Time Calculation:
The time taken for the missile to reach the aeroplane horizontally is given by:
[tex]\[ t = \frac{v}{a_{\text{horizontal}}} \][/tex]
Substituting the values:
[tex]\[ t = \frac{150}{40.95760221444959} = 3.6623237662843686 \, \text{seconds} \][/tex]
Thus, the time taken by the missile to reach the aeroplane is [tex]\(3.6623237662843686\)[/tex] seconds.
### Question (i): Sketch a vector diagram to represent the motion of the aeroplane and the missile
To visualize the motion, you should imagine the following diagram (a drawn version is recommended for clarity):
1. Aeroplane: Traveling horizontally to the right with a velocity of 150 m/s.
- Represent this as a horizontal arrow pointing to the right labeled 150 m/s.
2. Missile: Launched at an angle of [tex]\(35^\circ\)[/tex] above the horizontal ground with an acceleration of 50 m/s².
- Represent this as a diagonal arrow starting from the ground and angled [tex]\(35^\circ\)[/tex] from the horizontal axis.
- The arrow should show vectors for both the horizontal and vertical components of acceleration.
This diagram will display the vector components of both objects, emphasizing that the missile’s trajectory is straight towards the aeroplane.
### Other Requested Questions:
Feel free to provide the next three questions you would like to be solved. I am here to help you with them!
### Question (ii): Show that the time taken by the missile to reach the aeroplane is...
1. Given Data:
- Angle of missile launch, [tex]\(\theta = 35^\circ\)[/tex]
- Acceleration of missile, [tex]\(a_{\text{missile}} = 50 \, \text{m/s}^2\)[/tex]
- Velocity of the aeroplane, [tex]\(v_{\text{aeroplane}} = 150 \, \text{m/s}\)[/tex]
2. Convert Angle to Radians:
The angle in radians is given as:
[tex]\[ \theta_{\text{rad}} = 0.6108652381980153 \, \text{radians} \][/tex]
3. Horizontal Component of Missile’s Acceleration:
The horizontal component of the missile's acceleration is calculated using the formula:
[tex]\[ a_{\text{horizontal}} = a_{\text{missile}} \cdot \cos(\theta) \][/tex]
Substituting the values:
[tex]\[ a_{\text{horizontal}} = 50 \times \cos(0.6108652381980153) = 40.95760221444959 \, \text{m/s}^2 \][/tex]
4. Horizontal Velocity of the Missile:
To reach the aeroplane, the horizontal component of the missile's velocity must equal the aeroplane’s horizontal velocity:
[tex]\[ v_{\text{horizontal}\_\text{missile}} = 150 \, \text{m/s} \][/tex]
5. Time Calculation:
The time taken for the missile to reach the aeroplane horizontally is given by:
[tex]\[ t = \frac{v}{a_{\text{horizontal}}} \][/tex]
Substituting the values:
[tex]\[ t = \frac{150}{40.95760221444959} = 3.6623237662843686 \, \text{seconds} \][/tex]
Thus, the time taken by the missile to reach the aeroplane is [tex]\(3.6623237662843686\)[/tex] seconds.
### Question (i): Sketch a vector diagram to represent the motion of the aeroplane and the missile
To visualize the motion, you should imagine the following diagram (a drawn version is recommended for clarity):
1. Aeroplane: Traveling horizontally to the right with a velocity of 150 m/s.
- Represent this as a horizontal arrow pointing to the right labeled 150 m/s.
2. Missile: Launched at an angle of [tex]\(35^\circ\)[/tex] above the horizontal ground with an acceleration of 50 m/s².
- Represent this as a diagonal arrow starting from the ground and angled [tex]\(35^\circ\)[/tex] from the horizontal axis.
- The arrow should show vectors for both the horizontal and vertical components of acceleration.
This diagram will display the vector components of both objects, emphasizing that the missile’s trajectory is straight towards the aeroplane.
### Other Requested Questions:
Feel free to provide the next three questions you would like to be solved. I am here to help you with them!
