To simulate weightlessness for astronauts in training, the airplane needs to create a centrifugal force that cancels out the force of gravity. In this scenario, the centrifugal force at the top of the vertical circle should be equal in magnitude but opposite in direction to the force of gravity.
The centrifugal force is given by the equation:
F_c = m * v^2 / r
where F_c is the centrifugal force, m is the mass of the passengers, v is the velocity of the airplane, and r is the radius of the vertical circle.
At the top of the vertical circle, the centrifugal force should equal the force of gravity:
F_c = m * v^2 / r = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth).
Simplifying the equation, we can solve for the velocity (v) at the top of the vertical circle:
v^2 = r * g
v = sqrt(r * g)
Given that the radius of the vertical circle is 2.5 km (or 2,500 meters), and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the velocity at the top of the circle:
v = sqrt(2,500 * 9.8) = sqrt(24,500) ≈ 156.57 m/s
Therefore, the airplane should be moving at a velocity of approximately 156.57 m/s at the top of the vertical circle to simulate weightlessness for the passengers.