The “Vomit Comet.” In Zero-gravity astronaut training equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in Figure P4.47, the air-craf climbs from 24000ft to 31000ft,where it enters a parabolic path with a velocity of 143m/s at 45 degrees nose low. during this portion of the flight, the aircarft and objects inside its padded cabin are in free fall;astronauts and equipment float frely as if there were no gravity.what are the aircraft's (a)speed and (b) altitude at the top of the maneuver? (c) what is the time interval spent in microgravity?

To find the answers to these questions, we need to analyze the given information and apply relevant principles of physics.

(a) To find the speed of the aircraft at the top of the maneuver, we need to understand that the aircraft is climbing from 24,000ft to 31,000ft and then entering a parabolic path. Let's break it down step by step:

First, let's calculate the vertical distance the aircraft climbs: 31,000ft - 24,000ft = 7,000ft.

Next, we need to determine the time it takes for the aircraft to climb this distance. We can use the average vertical velocity of the aircraft during the climb.

The average vertical velocity can be found using the formula: v = d / t, where v is the vertical velocity, d is the vertical distance, and t is the time taken.

Assuming the climb is straight and level, we can estimate the time taken using the average velocity. Given that the velocity of the aircraft at 45 degrees nose low is 143m/s, we can use this value as the average vertical velocity.

So, the time taken to climb the 7,000ft can be calculated as: t = d / v = (7,000ft) / (143m/s) ≈ 48.95s.

Now, knowing the time it took for the climb, we can calculate the speed of the aircraft at the top of the maneuver. Since the aircraft is in free fall during this portion, its vertical velocity decreases due to gravity.

Using the principle of free fall, we know that the vertical velocity after the climb is zero at the top of the maneuver.

Therefore, the speed of the aircraft at the top of the maneuver is 0m/s.

(b) To find the altitude at the top of the maneuver, we need to consider the initial altitude of 31,000ft and the vertical distance climbed during the maneuver. Using the same calculation as in part (a), the aircraft climbs 7,000ft. Therefore, the altitude at the top of the maneuver is 31,000ft + 7,000ft = 38,000ft.

(c) To calculate the time interval spent in microgravity, we need to determine the duration of the parabolic path, where the astronauts and equipment experience free fall.

The time interval can be calculated using the principle of free fall and the formula: t = 2 * T, where T is the time it takes for a complete parabolic path.

To find T, we need to consider the properties of a parabolic trajectory. The portion of the parabola from the initial ascending climb to the top of the movement is symmetric to the portion from the top to the final descending path. Therefore, the time taken for each part is the same.

Since we already calculated the time taken for the initial climb as approximately 48.95s, we can multiply it by 2 to find the total time for a complete parabolic path. Thus, T = 2 * 48.95s ≈ 97.9s.

So, the time interval spent in microgravity is approximately 97.9s.

In summary:
(a) The speed of the aircraft at the top of the maneuver is 0m/s.
(b) The altitude at the top of the maneuver is 38,000ft.
(c) The time interval spent in microgravity is approximately 97.9s.