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Answer :
Final answer:
The rocket's original acceleration during the first 14 seconds of its flight was approximately 28.38 m/s², considering the total altitude reached after 22 seconds was 4989 meters.
Explanation:
To solve this question, correct use of the physics equations involving displacement, velocity, and acceleration is necessary. Initially, the rocket is launched with a certain acceleration. This acceleration lasts for 14 seconds, after which it stops, and the rocket continues its ascent due to inertia. The total time the rocket has been in the air when it reaches an altitude of 4989 meters is 22 seconds.
First, let's find the velocity of the rocket when the engine stopped after 14 seconds, we can denote this as v1. The equation of motion is v = u + at. Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Expanding this equation and plugging in our numbers gives v1 = 0 + a*14, as the rocket started initially at rest. So v1 = 14a.
The next phase is the free ascent lasting for 22-14 = 8 seconds. During this phase, it ascended vertically decelerating under the influence of gravity. Once the rocket engine cut off, the rocket continued to rise, but its speed gradually decreased due to the effect of gravity. Since we are ignoring air resistance, the only force acting on the rocket during this phase is the downward force of gravity, which decelerates the rocket at a rate of 9.8 m/s². Since the rocket is decelerating instead of accelerating, we can denote the acceleration due to gravity as -g.
Now, let's find the initial velocity (v0) and the final velocity (v) at t = 22s. The initial velocity was the velocity we calculated at t = 14s which was 14a. The final velocity was zero m/s. The rocket's ascend had momentarily stopped before it started descending. Thus we can write down the motion equation v = v0 - g*t: 0 = 14a - 9.8*8. Thus we have the second equation: 14a = 9.8*8.
The altitude at t = 22s was 4989m. The first 14 seconds it was powered by the rocket engine and ascended by s1 = a/2 * (14)^2 meters. The next 8 seconds it was moving due to momentum and ascended by s2 = (14a)*8 - (9.8)/2 * (8)^2 meters. (Here a is the original acceleration and 9.8 is the acceleration due to gravity.) Thus we have the third equation: s1 + s2 = 4989.
The solution of the equation system yields that the original acceleration of the rocket is approximately 28.38 m/s².
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