One of the methods used to train astronauts for the effects of "zero gravity" in space is to put them in a specially equipped plane which has been stripped of seats and fitted with padded walls. The pilot then takes the plane up to an altitude of typically 8700 meters and, at a speed of 900 km/hr, puts it into a sudden steep climb. He then pilots the plane so that it follows the parabolic trajectory of free fall until it is falling at the maximum allowable rate for safety. He can do this by flying the plane so that he is in continuous free fall himself. (He is trained to do this.) If the maximum angle of elevation that can be achieved in his initial rise is 18.0 degrees and likewise the maximum angle of descent for safety is 18.0 degrees, how long (in seconds) will the astronaut trainees experience zero gravity?

Question

One of the methods used to train astronauts for the effects of "zero gravity" in space is to put them in a specially equipped plane which has been stripped of seats and fitted with padded walls. The pilot then takes the plane up to an altitude of typically 8700 meters and, at a speed of 900 km/hr, puts it into a sudden steep climb. He then pilots the plane so that it follows the parabolic trajectory of free fall until it is falling at the maximum allowable rate for safety. He can do this by flying the plane so that he is in continuous free fall himself. (He is trained to do this.) If the maximum angle of elevation that can be achieved in his initial rise is 18.0 degrees and likewise the maximum angle of descent for safety is 18.0 degrees, how long (in seconds) will the astronaut trainees experience zero gravity?

Answer 1

To determine the duration of zero gravity experienced by the astronaut trainees, we can use the concept of free fall and the given information about the plane's trajectory.

Here are the steps to calculate the time of zero gravity:

1. Determine the total time for each parabolic trajectory:
- In each parabolic trajectory, the plane goes from the initial rise to the maximum angle of elevation, and then from the maximum angle of descent to the plane's return to level flight at the original altitude.
- Since the maximum angles of elevation and descent are both 18.0 degrees, the total angular range for each parabolic trajectory is 18.0 + 18.0 = 36.0 degrees.

2. Calculate the time spent in each parabolic trajectory:
- The time spent in each parabolic trajectory is determined by the angle covered and the rate of change of angle.
- The plane's angular velocity is determined by the speed and altitude.
- The angular velocity can be calculated using the formula: angular velocity = horizontal velocity / radius.
- In this case, the radius is the altitude of 8700 meters, and the horizontal velocity can be calculated using the speed of 900 km/hr.

3. Determine the total time of zero gravity:
- Since the plane follows a continuous parabolic trajectory, the total time of zero gravity is twice the time spent in each parabolic trajectory, as there are two trajectories per cycle.

Now, let's calculate the time of zero gravity:

Step 1: Convert the speed to meters per second:
- Speed = 900 km/hr = 900000 m/3600 s = 250 m/s.

Step 2: Calculate the angular velocity:
- Angular velocity = Horizontal velocity / Radius.
- Radius = 8700 meters.
- Horizontal velocity = Speed = 250 m/s.
- Angular velocity = 250 m/s / 8700 m = 0.0287 rad/s.

Step 3: Calculate the time spent in each parabolic trajectory:
- Time spent in each trajectory = Total angular range / Angular velocity.
- Total angular range = 36.0 degrees = 0.6283 radians (using the conversion factor: pi radians = 180 degrees).
- Time spent in each trajectory = 0.6283 radians / 0.0287 rad/s ≈ 21.850 seconds.

Step 4: Calculate the total time of zero gravity:
- Total time of zero gravity = 2 * Time spent in each trajectory.
- Total time of zero gravity = 2 * 21.850 s = 43.700 seconds.

Therefore, the astronaut trainees will experience zero gravity for approximately 43.700 seconds during each cycle of the maneuver.