Like this.
Not true.
When I was a kid I figured out how to get a skateboard moving from a dead stop, on flat ground, without putting a foot on the ground - and keep it moving after that. Here’s my shot at an explanation:
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Start with one foot on the rear of the board (so you’re able to kick up the front end), and the other somewhere forward on the board. Get your center of mass near your rear foot.
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Lean forward (your-forward, not board-forward), so you start to fall face-first. You are accelerating your center of mass sideways relative to the board, building up some velocity perpendicular to the direction the board is oriented.
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Now kick up the front end of the board and spin it 90 degrees to get it back under you so you stop falling. You and your board are now moving stably in a direction 90 degrees from the board’s original orientation.
This process can be repeated, falling alternately face-first and back-first, to build up speed. As you build up speed, the angle to which you re-orient the board each time gets less.
This is analogous to what a skater does. From a dead stop, they have to push off perpendicular to their rear blade. As they get going faster and faster, the kickoff angle of each individual pump action becomes smaller.