>you can easily verify that this works using basic mathematics and physics.
You are right, you absolutely can.
But such perpetuum mobile will not generate useful work (as in if you take away any energy from such engine, it will stop being infinite).
Plus, you have to have zero outside forces. So even tho the mentioned above engine is possible it is only possible EITHER way beyond the limits of our observable universe, where gravitational forces do not affect you (would unironically take tens of billions years to reach) OR at the centre of blackhole.
>What are the technological implications of having a perpetuum mobile?
Infinite energy heh. If it's small enough to fit in a car, it will completely destroy the economy around the world. Billions of people will go to powerty, millions will become insanely rich.
I simulated this with a friction coefficient of 25% and assumed that turning around a corner would also lose you 25% of the velocity. The result was still that the ball was propelled away after the turn further than it was pulled in before the turn.