In zero-gravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. The aircraft climbs from 24 000 ft to 31 000 ft, where it enters the zero-g parabola with a velocity of 143 m/s nose high at 45.0° and exists with velocity 143 m/s at 45.0° nose low. During this portion of the flight, the aircraft and objects inside its padded cabin are in free fall; they have gone ballistic. The aircraft then pulls out of the dive with an upward acceleration of 0.800g moving in a vertical circle with radius 4.13 km. (During this portion of the flight, occupants of the aircraft perceive an acceleration of 1.8g) What are the aircraft's (a) speed and (b) altitude at the top of the maneuver? (c) What is the time interval spent in zero gravity? (d) What is the speed of the aircraft at the bottom of the flight path?
i have done part a like this :
given:
(at begining of parabolic flight)
vx = 143 cos45 = 101 m/s
vy = 143 sin 45 = 101 m/s
(at end of parabolic flight)
v'x = 143 cos(-45) = 101 m/s
v'y = 143 sin (-45) =-101 m/s
The velocity of the aircraft at the top of the maneuver is the same as its horzontal component vx, that is 101 m/s
i don't know how to do the rest and i'm not even sure if part a) is right. please help! and please explain reasoning and why you chose certain equations and stuff. thank you!
To find the speed and altitude at the top of the maneuver, you can use the basic principles of physics, specifically the laws of motion and the concept of centripetal acceleration.
a) Speed at the top of the maneuver:
At the top of the maneuver, the aircraft is moving in a vertical circle with a radius of 4.13 km. This means that the centripetal force is acting towards the center of the circle. The upward acceleration during this portion of the flight is 0.800g, where g is the acceleration due to gravity.
To find the speed, you can use the centripetal acceleration formula:
ac = v^2 / r,
where ac is the centripetal acceleration, v is the velocity, and r is the radius of the circle.
Given that ac = 0.800g and r = 4.13 km, you can rearrange the formula to solve for v:
v = sqrt(ac * r).
Substituting the known values:
v = sqrt(0.800g * 4.13 km).
Note that it is essential to convert the radius from kilometers to meters to maintain consistent units. The gravitational acceleration, g, is approximately 9.8 m/s^2.
b) Altitude at the top of the maneuver:
To find the altitude at the top of the maneuver, you need to consider that the aircraft climbs from 24,000 ft to 31,000 ft initially.
The altitude at the top of the maneuver is the sum of the initial altitude (31,000 ft) and the change in altitude during the climb. The change in altitude is the difference between the initial altitude and the altitude at which the aircraft enters the zero-g portion of the flight (24,000 ft).
Thus, the altitude at the top of the maneuver is:
Altitude = 31,000 ft + (31,000 ft - 24,000 ft).
c) Time interval spent in zero gravity:
To find the time interval spent in zero gravity, you need to consider the parabolic flight path.
During the parabolic flight path, the aircraft and objects inside it are in free fall, experiencing a feeling of weightlessness. The time interval spent in zero gravity is equivalent to the time it takes for the aircraft to go from the nose high orientation to the nose low orientation.
This can be calculated using the formula for time of flight in projectile motion:
t = 2 * (v * sinθ) / g,
where t is the time of flight, v is the initial velocity, θ is the launch angle (45°), and g is the acceleration due to gravity.
d) Speed at the bottom of the flight path:
To find the speed at the bottom of the flight path, you can use the same principles as in part a) but with the downward acceleration.
Using the same centripetal acceleration formula:
ac = v^2 / r.
Given that ac = 1.8g (acceleration perceived by the occupants), you can solve for v using the radius of the vertical circle.
Remember to convert any necessary units and use the appropriate value for the acceleration due to gravity.
By following these steps, you should be able to solve parts b), c), and d) of the given problem.