Feeling "Weightless" When You Go "Over the Hump"
The phenomenon of "weightlessness" occurs when there is no force of support on your body. When your body is effectively in "free fall", accelerating downward at the acceleration of gravity, then you are not being supported. The sensation of apparent weight comes from the support that you feel from the floor, from the seat, etc. Different sensations of apparent weight can occur on a roller-coaster or in an aircraft because they can accelerate either upward or downward.
If you travel in a curved path in a vertical plane, then when you go over the top on such a path, there is necessarily a downward acceleration. Taking the example of the roller-coaster which is constrained to follow a track, then the condition for weightlessness is met when the downward acceleration of your seat is equal to the acceleration of gravity. Considering the path of the roller-coaster to be a segment of a circle so that it can be related to the centripetal acceleration, the condition for weightlessness is
The "weightlessness" you may feel in an aircraft occurs any time the aircraft is accelerating downward with acceleration 1g. It is possible to experience weightlessness for a considerable length of time by turning the nose of the craft upward and cutting power so that it travels in a ballistic trajectory. A ballistic trajectory is the common type of trajectory you get by throwing a rock or a baseball, neglecting air friction. At every point on the trajectory, the acceleration is equal to g downward since there is no support. A considerable amount of experimentation has been done with such ballistic trajectories to practice for orbital missions where you experience weightlessness all the time.
Standard mechanics problems