Well, here goes. This https://arxiv.org/pdf/physics/0210033.pdf has the equation for a falling chimney (e.g. the ones that crack apart on the way down) and the angular acceleration is proportional to sine of the angle from vertical. Oddly, the acceleration is inversely proportional to the length of the log but independent of its weight or density. They then say that the angular velocity is proportional to (1-cos (angle from vertical)) with the same odd independence as the acceleration. Once you have angular velocity (vs angle of top/log from vertical) cos of that angle is the horizontal component so you multiply cos of angle x angular velocity for the different angles as it falls. turns out the maximum occurs at 55 to 65 degrees from vertical - I calculated at 5 degree intervals so low res answer. Now this is based on gravity propulsion, no pull rope, no air damping slowing the swing e.g. a log (like the chimney) so it seems about right. So close the log notch at 60 degrees to pop it free. If you change the angular velocity profile the math will find the peak of the curve at a different angle. Just multiply cos angle x angular velocity at angle. The whole principle is that the further it swings down the faster it goes, but the less of that speed is horizontal component - i.e. the part that creates "throw".
An air damped branch laden top probably has a different velocity profile maybe with less pickup at the later angles as the air (square law for wind resistance) has more effect. That would probably raise the angle to 55 degrees or something less than the 60 degree log answer because the cos effect would be more dominant than the lessened magnitude of the angular velocity.
No p-toing has been accounted for. The first half of p-toing is shoving the spar back, taking away throw velocity. If it pops real early you've lost throw velocity and started from a backed up position. If it pops free with the spar sprung back but stopped, you've lost no throw velocity but lost some start position. If the spar sprang back to center as it pops free you've lost no start position and gained some throw velocity. etc. worst case the top is tugging the stem forward for a while before it frees. Honestly, with factors like the rigidity, mass of the spar, length etc a guy would be hard pressed to figure out if he added throw velocity or the top tugged you along if you thought you saw an ideal launch while riding it. There's a physics solution that shows pushing and pulling both occur when a "chimney" falls not on a pivot/hinge but on a friction surface where motion can occur besides pivoting. My knowledge absorption pulled a Homer Simpson when I found that stuff.
The next answer is what if you boost the initial angular velocity and then let gravity do the rest. Bit different equation for the angular velocity curve. Intuitively, the early part of the curve would have higher velocity values ratio-wise than the later angles like 70 degrees, and would interact better with multiplying times cos, so it would make a peak horizontal velocity component occur sooner than the 60 degrees log answer, perhaps substantially earlier. For another time ...
An air damped branch laden top probably has a different velocity profile maybe with less pickup at the later angles as the air (square law for wind resistance) has more effect. That would probably raise the angle to 55 degrees or something less than the 60 degree log answer because the cos effect would be more dominant than the lessened magnitude of the angular velocity.
No p-toing has been accounted for. The first half of p-toing is shoving the spar back, taking away throw velocity. If it pops real early you've lost throw velocity and started from a backed up position. If it pops free with the spar sprung back but stopped, you've lost no throw velocity but lost some start position. If the spar sprang back to center as it pops free you've lost no start position and gained some throw velocity. etc. worst case the top is tugging the stem forward for a while before it frees. Honestly, with factors like the rigidity, mass of the spar, length etc a guy would be hard pressed to figure out if he added throw velocity or the top tugged you along if you thought you saw an ideal launch while riding it. There's a physics solution that shows pushing and pulling both occur when a "chimney" falls not on a pivot/hinge but on a friction surface where motion can occur besides pivoting. My knowledge absorption pulled a Homer Simpson when I found that stuff.
The next answer is what if you boost the initial angular velocity and then let gravity do the rest. Bit different equation for the angular velocity curve. Intuitively, the early part of the curve would have higher velocity values ratio-wise than the later angles like 70 degrees, and would interact better with multiplying times cos, so it would make a peak horizontal velocity component occur sooner than the 60 degrees log answer, perhaps substantially earlier. For another time ...
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I didn’t just use my very first smiling face ever on any old joe smo, bro.
