No math required in this case. A speed difference between hand and spool does not imply that there's any acceleration happening. Zooterkin was right, you were wrong. If a=0, it does *not* mean the spool is stationary, as you claimed, just that its speed isn't changing. Sir Isaac worked this out centuries ago.
How more basic can your errors become?
An opposing force F at the ground, becomes 2F at the thread,because of the gearing. 2F - F =F. It can't stop accelerating. What actually happens, is dependent upon the ratio of the moment of the reel to the mass of the reel.
But, in the simplified case, acceleration stops when F net = 0, but the gearing simply prevents that. To keep a constant velocity, the force must be applied for half the time. One way is to not pull with a constant force as you watch the reel respond, and the other is to let it pass by so you can't apply force until you catch up with it, and pull again. The result is the same. Force is applied for half the time. V_avge reel = V_avge hand.
The cart is based on discredited notions of screwing props and cotton reels.
Exhumed, to repeat the mistakes of the past.
The first source was Ruina and his wheels and rails, just like the ruler cart. Then Drela. Spork's analysis was a version of that, with the "arbitrarily low" friction of .001. Steel on steel.
Now it's Smith, and the air lever....etc
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