Principles of Magical Aerodynamics
I. Underlying Principles:
The non expert tends to think of magic as transcending the laws of physics. That's mainly a misconception. In particular, Newton's laws of motion retain their validity. For broomstick rider as well as bird or helicopter, staying aloft requires a non-inertial trajectory (steady upward acceleration)in the Earth's gravitational field, and consequently a force (Newton's Second Law). A steady flow of momentum relative to the inertial frame is needed. This momentum flow comes almost entirely in the form of the reaction force of air (Newton's third law), so an equal flow of momentum goes into accelerating air downward.
For a rider of mass m a force mg is required to hold the rider up, where g is the acceleration due to gravity. This force is balanced by changing the momentum p of air at a rate dp/dt = mg. If we simplify by assuming that a mass M of air has its velocity changed at a rate of dv/dt, then Newton's third law gives M(dv/dt) = mg. For a 60 kg broom rider, mg = 588 N, which would correspond to changing the velocity of 588 kg of air by 1 m/s every second. At 1.2 kg/m^3 of air, that amounts to 490 m^3 of air - roughly the volume of a suburban house. Keeping a single light weight rider aloft thus requires throwing down a houseful of air every second at 1 m/s or every 10 seconds at 10 m/s. No doubt you have noticed the blast of air as a broomstick rider flies closely by.
The same principles apply to a 1 kg owl, 3000 kg helicopter, and a 200,000 kg 747 jetliner, with dp/dt proportional to the mass.
What about lighter than air vehicles, such as helium balloons or flying carpets? They too must manage the momentum transfer trick, though they do it a bit differently. Every second, zillions of air molecules moving rather smartly (at about 150% of the speed of sound, on average) smash into us from all directions and zip off at roughly the same speed away from us, thus changing their momentums drastically. Since they are coming from all directions, the forces due to these momentum changes tend to cancel out.
We can divide the forces on the bubble of air that supports a flying carpet into three parts, First, those due to air collisions from the side, which cancel out since each force is balanced by an equal force from the opposite direction. Second, the weight of the carpet, its occupants, and the air in its "bubble." Third, the forces from collisions from above and collisions from below. The third category of forces do not balance, since the air pressure (=force due to collisions per unit area) decreases with height. The net force due to that pressure change is exactly the weight of the air that would occupy the bubble at normal density. The carpet operates by decreasing the density of the air in its bubble - that decrease is responsible for the slight "pop" you feel in your ears as the carpet lifts off. A similar mechanism is responsible for the lift in a hot air balloon.
The non expert tends to think of magic as transcending the laws of physics. That's mainly a misconception. In particular, Newton's laws of motion retain their validity. For broomstick rider as well as bird or helicopter, staying aloft requires a non-inertial trajectory (steady upward acceleration)in the Earth's gravitational field, and consequently a force (Newton's Second Law). A steady flow of momentum relative to the inertial frame is needed. This momentum flow comes almost entirely in the form of the reaction force of air (Newton's third law), so an equal flow of momentum goes into accelerating air downward.
For a rider of mass m a force mg is required to hold the rider up, where g is the acceleration due to gravity. This force is balanced by changing the momentum p of air at a rate dp/dt = mg. If we simplify by assuming that a mass M of air has its velocity changed at a rate of dv/dt, then Newton's third law gives M(dv/dt) = mg. For a 60 kg broom rider, mg = 588 N, which would correspond to changing the velocity of 588 kg of air by 1 m/s every second. At 1.2 kg/m^3 of air, that amounts to 490 m^3 of air - roughly the volume of a suburban house. Keeping a single light weight rider aloft thus requires throwing down a houseful of air every second at 1 m/s or every 10 seconds at 10 m/s. No doubt you have noticed the blast of air as a broomstick rider flies closely by.
The same principles apply to a 1 kg owl, 3000 kg helicopter, and a 200,000 kg 747 jetliner, with dp/dt proportional to the mass.
What about lighter than air vehicles, such as helium balloons or flying carpets? They too must manage the momentum transfer trick, though they do it a bit differently. Every second, zillions of air molecules moving rather smartly (at about 150% of the speed of sound, on average) smash into us from all directions and zip off at roughly the same speed away from us, thus changing their momentums drastically. Since they are coming from all directions, the forces due to these momentum changes tend to cancel out.
We can divide the forces on the bubble of air that supports a flying carpet into three parts, First, those due to air collisions from the side, which cancel out since each force is balanced by an equal force from the opposite direction. Second, the weight of the carpet, its occupants, and the air in its "bubble." Third, the forces from collisions from above and collisions from below. The third category of forces do not balance, since the air pressure (=force due to collisions per unit area) decreases with height. The net force due to that pressure change is exactly the weight of the air that would occupy the bubble at normal density. The carpet operates by decreasing the density of the air in its bubble - that decrease is responsible for the slight "pop" you feel in your ears as the carpet lifts off. A similar mechanism is responsible for the lift in a hot air balloon.
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