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Sagot :
Sure, let's break the problem down step-by-step:
### (a) Maximum Height Reached
To calculate the maximum height reached by the tennis ball, we can use the kinematic equation for vertical motion under uniform acceleration:
[tex]\[ h = \frac{v^2}{2g} \][/tex]
where:
- [tex]\( v \)[/tex] is the initial velocity of the ball (7 m/s)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s²)
By plugging in the values:
[tex]\[ h = \frac{7^2}{2 \times 9.81} \][/tex]
The calculated maximum height [tex]\( h \)[/tex] is approximately [tex]\( 2.497 \, \text{meters} \)[/tex].
### (b) Time Taken to Reach the Maximum Height
The time taken to reach the maximum height can be determined using the formula:
[tex]\[ t = \frac{v}{g} \][/tex]
where:
- [tex]\( v \)[/tex] is the initial velocity (7 m/s)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s²)
By substituting the values into the formula:
[tex]\[ t = \frac{7}{9.81} \][/tex]
The time to reach maximum height [tex]\( t \)[/tex] is approximately [tex]\( 0.714 \, \text{seconds} \)[/tex].
### (c) Total Time of Flight
For the total time of flight, consider that the time to rise to the maximum height is equal to the time to fall back to the starting point. Therefore:
[tex]\[ \text{Total Time of Flight} = 2 \times t \][/tex]
By using the time [tex]\( t \)[/tex] calculated in part (b):
[tex]\[ \text{Total Time of Flight} = 2 \times 0.714 \][/tex]
The total time of flight is [tex]\( 1.427 \, \text{seconds} \)[/tex].
### (d) Parameters to Consider for Free Falling Objects
When analyzing the motion of free falling objects, the following parameters are generally considered:
1. Initial Velocity (u): The starting velocity of the object, in this case, 7 m/s.
2. Gravity (g): The acceleration due to gravity, which on Earth is approximately 9.81 m/s².
3. Time of Flight (t): The total duration the object is in motion, in this case, 1.427 seconds.
4. Height (h): The displacement or maximum height reached by the object, here it is approximately 2.497 meters.
5. Final Velocity (v): The velocity of the object just before it returns to its starting point. (For free fall, the final velocity at the lowest point would consider speed at impact/return).
These parameters are essential for solving problems involving the motion of objects under the influence of gravity.
The final numerical values considering these principles are:
- Maximum Height: [tex]\( 2.497 \, \text{meters} \)[/tex]
- Time to Maximum Height: [tex]\( 0.714 \, \text{seconds} \)[/tex]
- Total Time of Flight: [tex]\( 1.427 \, \text{seconds} \)[/tex]
- Parameters to Consider: Initial Velocity, Gravity, Time of Flight, Height, Final Velocity
### (a) Maximum Height Reached
To calculate the maximum height reached by the tennis ball, we can use the kinematic equation for vertical motion under uniform acceleration:
[tex]\[ h = \frac{v^2}{2g} \][/tex]
where:
- [tex]\( v \)[/tex] is the initial velocity of the ball (7 m/s)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s²)
By plugging in the values:
[tex]\[ h = \frac{7^2}{2 \times 9.81} \][/tex]
The calculated maximum height [tex]\( h \)[/tex] is approximately [tex]\( 2.497 \, \text{meters} \)[/tex].
### (b) Time Taken to Reach the Maximum Height
The time taken to reach the maximum height can be determined using the formula:
[tex]\[ t = \frac{v}{g} \][/tex]
where:
- [tex]\( v \)[/tex] is the initial velocity (7 m/s)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s²)
By substituting the values into the formula:
[tex]\[ t = \frac{7}{9.81} \][/tex]
The time to reach maximum height [tex]\( t \)[/tex] is approximately [tex]\( 0.714 \, \text{seconds} \)[/tex].
### (c) Total Time of Flight
For the total time of flight, consider that the time to rise to the maximum height is equal to the time to fall back to the starting point. Therefore:
[tex]\[ \text{Total Time of Flight} = 2 \times t \][/tex]
By using the time [tex]\( t \)[/tex] calculated in part (b):
[tex]\[ \text{Total Time of Flight} = 2 \times 0.714 \][/tex]
The total time of flight is [tex]\( 1.427 \, \text{seconds} \)[/tex].
### (d) Parameters to Consider for Free Falling Objects
When analyzing the motion of free falling objects, the following parameters are generally considered:
1. Initial Velocity (u): The starting velocity of the object, in this case, 7 m/s.
2. Gravity (g): The acceleration due to gravity, which on Earth is approximately 9.81 m/s².
3. Time of Flight (t): The total duration the object is in motion, in this case, 1.427 seconds.
4. Height (h): The displacement or maximum height reached by the object, here it is approximately 2.497 meters.
5. Final Velocity (v): The velocity of the object just before it returns to its starting point. (For free fall, the final velocity at the lowest point would consider speed at impact/return).
These parameters are essential for solving problems involving the motion of objects under the influence of gravity.
The final numerical values considering these principles are:
- Maximum Height: [tex]\( 2.497 \, \text{meters} \)[/tex]
- Time to Maximum Height: [tex]\( 0.714 \, \text{seconds} \)[/tex]
- Total Time of Flight: [tex]\( 1.427 \, \text{seconds} \)[/tex]
- Parameters to Consider: Initial Velocity, Gravity, Time of Flight, Height, Final Velocity
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