Final answer:
To calculate the time a bottle rocket takes to reach the peak of its ascent, use the kinematic equation, considering acceleration until the fuel runs out and then deceleration due to gravity to find total ascent time.
Step-by-step explanation:
The question involves calculating the time taken for a bottle rocket to reach its maximum height when subjected to an initial acceleration and then affected by gravity. Using the kinematic equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can determine the time taken to reach the point where the engines shut off and the rocket reaches a height of 200 m. Since the final velocity at this point will be 0 m/s (the peak of its ascent), we can rearrange the formula to find t: t = (v - u) / a. Ignoring air resistance and assuming the rocket accelerates until it runs out of fuel, the total time to reach the maximum height will be the time to reach 200 m plus the time it takes for gravity to decelerate the rocket from its final velocity at 200 m to 0 m/s at its peak. This calculation involves balancing the kinematic equations for acceleration and deceleration under gravity.
For the other questions posed, if air resistance is ignored, the rocket would reach approximately 46 m in height. However, a lesser height implies there might be additional factors like air resistance or possible design inefficiencies affecting its ascent. Regarding conservative forces and potential/kinetic energy changes: if a particle is affected solely by conservative forces, the total mechanical energy (potential energy + kinetic energy) of the particle would remain constant; however, the individual kinetic and potential energies could vary during the trip. For a rocket releasing its booster at a given height and velocity, the maximum height, and speed post-release can be estimated using energy conservation principles. Lastly, the initial mass of a rocket can be deduced from its acceleration and fuel characteristics.