Using our world's physics, how could one explain the mechanics behind a witch on a broomstick? Not just the propelling part, but even the most basic fact - sitting on a broomstick - seems problematic: The center of gravity of human body is far above the broomstick, so within a second, a witch would just turn flying upside down (making landings very painful among other problems).
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Let's assume witches do exist and they are riding broomstick-like objects!
My answer could be decomposed into multiple points:
How would they mount the broomstick without hanging from the broomstick?
I think a well-behaved witch (if there are any) would never mount the broomstick like man, but like ladies of her time. In this case balancing on the stick could be learned (we all have set on climbing frame rods at young ages without falling).
An evil and promiscuous witch might ride a broomstick like man ride their horses, but I think a witch like that would be practical enough to apply a cross-rod or some sort of a saddle. I mean we only see witches from quite far, and their dress hides the fine details. A practical witch might also apply weighted boots or other counterweights to maintain upward position.
It is also worth mentioning, that medieval-time broomsticks were not manufactured, but handcrafted so they could be directly shaped for riding like in Harry Potter movies.
How could a broomstick-like object fly?
The most appealing explanation would be a rocket- or jet drive, which could - in theory - be built into the body of the broomstick, and exhaust gases could be released by shafts through the sorghum. The only problem with this explanation would be, that this way the broomstick would be propelled forward with an extremely high power, so it would be very fast, hard to control and very hard to hang on.
If I had to build something like that I would choose another approach:
I would use a flying machine of some sort (the "what sort" doesn't really matter in our point of view), and hide it with some sort of stealth tech. (Active camouflage of some sort for example) and fix the brooms ends to it. The witch would ride the broom and control the flying machine remotely .
Traditionally witches ride them straw side forward like a hobby-horse.
The posture in the first picture seems the best way to ride a broom. By sitting on the straw and keeping the handle up lowers the center of gravity and prevents tipping over. Unfortunately, it is not very aerodynamic as your upright posture catches the wind. Go to fast and you get blown right off the things. You can always trade speed for stability by leaning forward with you head close to the handle knees bent feet close to the straw. but don't blame me when you barrel roll and lose your grip.
I think by controlling your posture carefully for the task at hand, you could ride a broom with some degree of realism.
Logically, witches are made of wood and weigh as much as a duck!
So there isn't much center of mass to deal with!
On a slightly more serious note, witches are already using magic to fly around on brooms in the sky. So it probably does't take much more magic to ensure they stay upright, especially after the first embarrassing landing, with their skirts over their head.
A recent anime (and ongoing manga) series Flying Witch provided an alternate explanation.
After the main character complained that she could not 'ride' a broom for too long before her butt hurt and her legs got tired, her older sister explained that she was not supposed to lift her body up with the broom, but rather transfer the levitation from the broom back into her self. She was then made to practice by holding the broom vertically with one hand, and hovering both herself and the broom a foot or so above the ground.
The 'riding' position, is for increased contact with the broom, for efficiency of transfer and safety. And it provides slightly better aerodynamics than standing on it or hanging from it. Ladylike witches ride sidesaddle, because it is more refined. But athletic witches mount the broom so they can maximize contact and minimize drag, and thus get maximum speed and control.