In zero-gravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in the figure, the aircraft climbs from 24,000 ft to 32000 ft, where it enters the zero-g parabola with a velocity of 158 m/s at 45.0° nose high and exits with velocity 158 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 4.06 km. (During this portion of the flight, occupants of the plane perceive an acceleration of 1.800g.)

Question

In zero-gravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in the figure, the aircraft climbs from 24,000 ft to 32000 ft, where it enters the zero-g parabola with a velocity of 158 m/s at 45.0° nose high and exits with velocity 158 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 4.06 km. (During this portion of the flight, occupants of the plane perceive an acceleration of 1.800g.)

What is the aircraft's altitude at the top of the maneuver?

Answer 1

To find the aircraft's altitude at the top of the maneuver, we can use the equations of motion for vertical circular motion.

Given:
Initial velocity, u = 158 m/s
Angle of trajectory, θ = 45.0°
Radius of the circular path, r = 4.06 km = 4060 m
Upward acceleration, a = 0.800g = 0.800 * 9.8 m/s² (g is the acceleration due to gravity)

Step 1: Finding the time taken to reach the top of the maneuver
The initial velocity can be resolved into horizontal and vertical components:
u_horizontal = u * cos(θ)
u_vertical = u * sin(θ)

At the topmost point, the vertical velocity becomes zero, and we can use the following equation:
v_vertical = u_vertical + a * t
0 = u_vertical + a * t

Substituting the values:
0 = u * sin(θ) + a * t

Solving for t:
t = -u * sin(θ) / a

Step 2: Calculating the displacement at the top of the maneuver
The displacement of an object in vertical circular motion can be calculated using the equation:
s = v_initial * t + (1/2) * a * t²

Substituting the values:
s = u_vertical * t + (1/2) * a * t²

Since u_vertical = u * sin(θ), the equation becomes:
s = u * sin(θ) * t + (1/2) * a * t²

Step 3: Finding the altitude at the top of the maneuver
The altitude at the top of the maneuver is equal to the sum of the initial altitude and the displacement.

Given:
Initial altitude, h_initial = 24000 ft = 7315.2 m

Altitude at the top = h_initial + s

Substituting the values, we can now calculate the altitude at the top of the maneuver.