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 8600 meters and, at a speed of 870 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 21.5 degrees (with respect to the horizontal) and likewise the maximum angle of descent for safety is 21.5 degrees, how long (in seconds) will the astronaut trainees experience zero gravity?
To determine how long the astronaut trainees will experience zero gravity, we need to calculate the time it takes for the plane to follow the parabolic trajectory of free fall.
First, let's consider the ascent phase of the plane's flight. We know that the maximum angle of elevation that can be achieved is 21.5 degrees.
The plane's velocity in the ascent phase can be determined using trigonometry. We can use the horizontal component of the plane's velocity, which is equal to the ground speed of 870 km/hr.
Velocity at ascent phase (horizontal component) = 870 km/hr * cos(21.5 degrees)
Next, we need to determine the time it takes for the plane to reach the maximum altitude, which is typically 8600 meters. We can use the kinematic equation:
y = v0*t + (1/2)*a*t^2
In this equation, y represents the maximum altitude, v0 represents the initial velocity (which is the velocity at the ascent phase), a represents the acceleration (which is equal to the acceleration due to gravity), and t represents time.
First, let's convert the velocity from km/hr to m/s:
Velocity at ascent phase (horizontal component) = 870 km/hr * (1000 m/1 km) * (1 hr/3600 s)
Now we can plug in the values into the kinematic equation:
8600 m = (Velocity at ascent phase) * t + (1/2) * (-9.8 m/s^2) * t^2
Simplifying the equation, we get:
-4.9 t^2 + (Velocity at ascent phase) * t - 8600 = 0
Solving this quadratic equation using the quadratic formula, we can find the time it takes for the plane to reach the maximum altitude.
t = (-b ± √(b^2 - 4ac)) / 2a
where a = -4.9, b = (Velocity at ascent phase), and c = -8600
Substituting the values, we can solve for t.
Now, let's consider the descent phase of the plane's flight. We know that the maximum angle of descent for safety is also 21.5 degrees.
Using the same approach as before, we can determine the velocity at the descent phase:
Velocity at descent phase (horizontal component) = 870 km/hr * cos(21.5 degrees)
Next, we need to determine the time it takes for the plane to descend from the maximum altitude to the initial altitude. Again, we can use the kinematic equation:
y = v0*t + (1/2)*a*t^2
In this case, y represents the difference in altitude between the maximum and initial altitudes, v0 represents the initial velocity (which is the velocity at the descent phase), a represents the acceleration (which is equal to the acceleration due to gravity), and t represents time.
Using the same process as before, we can solve for t.
Finally, we can calculate the total time the astronaut trainees will experience zero gravity by adding the time of the ascent phase and the time of the descent phase.
Total time = time of ascent phase + time of descent phase
To determine the duration of zero gravity experienced by the astronaut trainees, we need to calculate the time it takes for the plane to follow the parabolic trajectory during its ascent and descent.
First, let's consider the ascent phase. The maximum angle of elevation is given as 21.5 degrees. At this angle, the plane is flying in a parabolic trajectory, and its vertical acceleration can be assumed to be equal to the acceleration due to gravity (9.8 m/s^2) during this phase.
We can use the kinematic equation for vertical motion:
s = ut + (1/2)gt^2
Where:
- s is the vertical displacement (which will be zero for the ascent phase since the plane starts and ends at the same height)
- u is the initial vertical velocity (zero since the plane is initially at rest)
- g is the acceleration due to gravity
- t is the time
Using this equation, we can solve for the time taken for the ascent phase:
0 = (1/2)(9.8)t^2
0 = 4.9t^2
Since the coefficient and acceleration are both positive, we can ignore the negative solution. Solving for t gives us:
t = sqrt(0/4.9) = 0
This means that during the ascent, the trainees experience zero gravity instantaneously.
Now let's consider the descent phase. The maximum angle of descent is also given as 21.5 degrees. Using the same reasoning as before, we know the vertical acceleration during the descent phase will also be equal to the acceleration due to gravity (9.8 m/s^2).
Using the same kinematic equation as before, but now with a vertical displacement of zero during the descent phase, we can solve for t again:
0 = (1/2)(9.8)t^2
0 = 4.9t^2
Again, we can ignore the negative solution. Solving for t gives us:
t = sqrt(0/4.9) = 0
This means that during the descent as well, the trainees experience zero gravity instantaneously.
Therefore, the total time that the astronaut trainees experience zero gravity is 0 seconds.