A Catapult Launches A Test Rocket Vertically Upward From A W
A Catapult Launches A Test Rocket Vertically Upward From A Well Givin
A catapult launches a test rocket vertically upward from a well, giving the rocket an initial speed of 80.2 m/s at ground level. The engines then fire, and the rocket accelerates upward at 3.80 m/s² until it reaches an altitude of 1190 m. At that point, its engines fail, and the rocket goes into free fall, with an acceleration of -9.80 m/s². The motion during engine operation and free fall must be considered separately to analyze the rocket's trajectory fully.
Paper For Above instruction
The problem involves analyzing a vertical launch scenario of a rocket from a well, considering different phases of motion: powered flight under engine acceleration and free fall after engine failure. The key to solving this problem lies in decomposing the motion into these phases and applying kinematic equations accordingly.
Phase 1: Initial Launch and Powered Ascent
Initially, the rocket receives an initial velocity of 80.2 m/s from the catapult. During this phase, the rocket's acceleration results from the engines firing, providing a constant acceleration of 3.80 m/s² until it reaches an altitude of 1190 meters. To analyze this phase, we use the kinematic equation:
- v = v₀ + a·t
- s = v₀·t + 0.5·a·t²
where v is the final velocity at altitude 1190 meters, v₀ is the initial velocity (80.2 m/s), a is the acceleration (3.80 m/s²), t is the time during powered ascent, and s is the displacement during this phase.
Calculating Final Velocity at Engine Failure
To find the velocity of the rocket at 1190 meters when the engines cut off, we first need to determine the time taken to reach this altitude during powered ascent. However, since the initial velocity and acceleration are known, and the displacement is known, we can directly compute the velocity at that point using the equation:
v² = v₀² + 2·a·s
where:
- v₁ = velocity at 1190 meters after powered ascent
Substituting the known values:
v₁² = (80.2 m/s)² + 2·(3.80 m/s²)·(1190 m)
v₁² = 6432.04 + 2·3.80·1190
v₁² = 6432.04 + 9024
v₁² = 15456.04
v₁ = √15456.04 ≈ 124.3 m/s
This velocity (~124.3 m/s) is just before engine failure at 1190 meters.
Phase 2: Free Fall After Engine Failure
After the engines fail at 1190 meters, the rocket continues upward until it reaches its maximum height, then begins to descend. During free fall, the acceleration is -9.80 m/s², and the initial velocity for this phase is v₁ (~124.3 m/s). We analyze the ascent to maximum height and subsequent descent using the same kinematic equations.
Determining Maximum Height
The maximum height is reached when the velocity reduces to zero during upward motion under gravity. Using:
v² = v₁² + 2·a·Δs
with v = 0 (at maximum height), v₁ = 124.3 m/s, a = -9.80 m/s², and Δs = h_max - 1190 m (additional height gained after engine failure).
Solving for Δs:
0 = (124.3)² + 2·(-9.80)·Δs
2·9.80·Δs = (124.3)²
Δs = (124.3)² / (2·9.80) = 15456.04 / 19.6 ≈ 788.2 m
This is the additional altitude gained after engine failure, so the maximum height is:
h_max = 1190 m + 788.2 m ≈ 1978.2 m
Descent Phase
After reaching maximum height at approximately 1978.2 meters, the rocket starts descending, accelerating downward at 9.80 m/s² (ignoring air resistance). Its initial velocity at this point is zero (at the apex). To find the time to reach the ground after engine failure, we analyze the free fall from this maximum height, considering initial velocity zero:
s = 0.5·g·t²
and solving for t:
t = √(2·h / g) = √(2·1978.2 / 9.80) ≈ √(403.7) ≈ 20.1 seconds
Thus, it takes approximately 20.1 seconds from maximum height to impact with the ground, not accounting for initial downward velocity since it is zero at the maximum height.
Summary of the Motion
The entire ascent includes: initial launch, powered ascent to 1190 meters, free-fall ascent to approximately 1978.2 meters, and then descent back to the ground. The maximum height reached is approximately 1978.2 meters, and the total duration from launch to impact involves calculating times during each phase.
Conclusion
This analysis demonstrates the importance of breaking down rocket motion into phases when dealing with varying acceleration. Understanding the initial launch parameters, applying kinematic equations, and considering gravity's influence are fundamental in predicting the trajectory and maximum height of a rocket launched under specified conditions. The complex interplay of initial velocities, engine acceleration, and gravitational pull underscores the significance of precise calculations in aerospace engineering and trajectory planning.
References
- Serway, R. A., & Jewett, J. W. (2018). Fundamentals of Physics (11th ed.). Cengage Learning.
- Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers (6th Ed.). W. H. Freeman and Company.
- Walker, J. S. (2014). Physics (4th ed.). Pearson.