This is not the right place for technicalities, but here is some physics, something which can be useful sometime:
Hauling is an effective mechanism to increase line speed. The straightforward explanation coming to our mind is that if we add speed to (line) speed, it must be effective. If this was the only phenomenon involved, then the most desirable timing for peak haul velocity (PHV) would be just before rod straight position, when tip speed is maximal. There would not be any side effect, the line sliding inside the guides, so we can wonder why one can get a tailing loop when one hauls the line.
To identify the cause of a tail in the line, in other words the cause of a drop in the tip path leading to that tail, we need Mr. Newton’s help. I shall keep things as simple as possible to ease the understanding of the phenomenon, so there can be some shortcut. Without hauling, the force at tip pulling on the line (line mass * line acceleration) equals the force deflecting the (spring of) the rod (rod stiffness * rod deflection). We create energy in the line by moving that force over the distance covered by the tip of the rod. Thus a good idea is to increase the distance along which the force acting on the line is moved, and this is what we do with the haul. In practice, the line outside the guides is shortened (by the haul distance) and its speed is increased, and then Mr. Newton tells us that we have created another force on the spring: changing mass and or speed changes the momentum of the line (which equals the product of both), and corresponds to a force. The famous F=ma is a simplification of this first law by considering that m (mass) remains a constant. The increase in force is highest when the change in line momentum is highest, that is to say at the peak acceleration of the haul. And this peak acceleration always takes place during the first half of the haul, before PHV (for PHV the acceleration of the haul is nil).
So not only do we move a force over an increased distance, but we move a higher force over this longer distance. This is a kind of kiss cool effect. The increase in energy that we can produce like that can be pretty large (e.g. 50% more energy for the line) since we place the haul at a moment when the force on the spring is at its highest. However there is always a price to pay for that, and guess what, this is the risk for a tail in the fly leg.
Since we create an added force on the spring, the rod is bending a little bit further and if the rod butt is still back from vertical at this time, then the tip moves down from its trajectory. So we realize that in unfavorable conditions, we can get a tail and that this is a question of timing for the haul. The side effect of the haul is less visible on the tip path if the pull angle is beyond 90 degrees from the rod butt, the extra deflection being practically in line with the tip path. The conclusion is that it is better to haul late in the cast to minimize the risk of tail.
If we concentrate on this side effect of the haul, there is some logic to think that the extra force should follow the bending propensity of the rod coming from the cast itself. This synch within cast and haul maximizes the extra elastic energy storage due to the haul (e.g. by one third). If there is a difference in synch, then the rod gets two signals to bend down that might come at a different time and it either troubles the rate of bending of the rod or eventually the side effect of the haul could oppose the unloading of the rod if the haul comes very late. Bad luck indeed because the direct effect of the haul, the added speed, is always there and cannot easily be synchronized with the side effect of the haul: one is depending on PHV, the other one on peak haul acceleration, and they do not take place at the same time. Gods of mechanics are of no help here despite the delay in rod response; getting the best of both effects is limited to stiff/fast rods (e.g. from #10 or something), but variation in performance remain small as long as the haul takes place within the right window of time. This window can cover the time range between MAV and RSP for stiff/fast rods, but gets smaller and closer to MAV for soft/slow rods. One can “overhaul” a trout rod.
In terms of extra speed generated, the side effect is more effective than the direct effect, but the side effect brings the risk of tail in the fly leg. There is good news here; it means that there is some window for hauling: either you try to get the best from the direct effect; either you try to get the best from the side effect with a risk of tail. If PHV falls within maximum angular velocity and rod straight position, e.g. right in the middle, then this is a fair compromise. Bad luck again because it appears easier to haul and rotate at the same time (MAV).
For a fisherman, hauling late is on the safe side of haul time tuning, maybe not the best speed but no tail to fear about, for the competition caster his interest is to come as close as possible to the “tail zone” and get the best of line speed. In that case the competition caster has to produce a higher torque to make the cast at a given pace (e.g. 25% more torque, but in practice the caster may react by limiting this torque increase). It may appear awkward to tune but some studies show that trained casters are able to adapt their haul timing to the casting conditions.
It is also wiser to use a speedy long distance haul than a brutal short haul. I trained myself hauling but I was highly influenced by the side effect of the haul which I could see in my experiments, and guess what, I have to suffer from a bad habit and get a tail in the fly leg quite often. The cure is to stop hauling short and use a long progressive haul. I really learned it wrong with a sharp haul close to maximum rod bend after reading a book on casting (maximum rod bend happens shortly after maximum angular velocity).
In terms of tackle, a faster/stiffer one can withstand a haul on the earlier side of timing (a stiffer rod is less prone to bending and is more resistant to the side effect), whilst a slower tackle needs a haul on the later side of the “hauling window”, it tends to drive a tail more easily. Hauling within the right window of time needs training for sure, and watching your cast can help tuning the haul with the cast.
Hope this is understandable
Merlin