Answer :
Final answer:
The rocket reaches a maximum height of 2250 meters after 15 seconds of powered flight, and it takes a total of 76.2 seconds to return to the ground. The rocket achieves its maximum velocity of 300 m/s at the end of the fuel burn. Therefore, the maximum height is at the peak of powered ascent.
Explanation:
Rocket Motion Analysis
A model rocket is launched vertically upwards with an initial acceleration due to thrust. In this scenario, we will calculate the maximum height the rocket reaches, the total time it stays in the air, and the point at which it achieves maximum velocity.
1. Maximum Height Calculation
The rocket accelerates at 20 m/s² for 15 seconds. Using the formula for distance under constant acceleration:
d = ut + (1/2)at²,
Where:
- d = distance (height)
- u = initial velocity = 0 m/s (starts from rest)
- a = acceleration = 20 m/s²
- t = time = 15 s
Substituting the values:
d = 0 15 + (1/2) 20 (15)² = 0 + 0.5 20 225 = 2250 meters.
2. Total Time in the Air
After burning fuel for 15 seconds, the rocket will continue to ascend until its velocity reaches zero. First, we need to calculate the velocity at the end of fuel burnout using:
v = u + at,
where:
- v = final velocity
- u = initial velocity = 0 m/s
- a = 20 m/s²
- t = 15 s
Thus, v = 0 + 20 15 = 300 m/s.
Now, we can find the time to reach maximum height using v = u + at, where the final velocity (v) at the peak is 0 m/s:
0 = 300 - 9.81t, so solving for t gives:
t = 300 / 9.81 (approximately 30.6 seconds).
The total time in the air will be the ascent (15 seconds fuel burn + 30.6 seconds coasting) and descent (30.6 seconds to fall back down), giving:
Total time = 15 + 30.6 + 30.6 = 76.2 seconds.
3. Maximum Velocity Position
The rocket reaches its maximum velocity at the end of the fuel burn after 15 seconds, where it achieves 300 m/s just before it starts to decelerate due to gravity.
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