Tom...
I think your logic is basically on target. I'm not so sure about your math.
Taking the math out of it, we have some principles to consider:
1.) When shooting line, it doesn't simply shoot out the loop with the same dynamics as those of the loop when not shooting.....many things change.
2.) Steve Rajeff has pointed out that, "overhang" (which we can consider, for practical purposes, the shooting line or running line behind any head and between the rear of the head and the rod tip) is, "the barometer for the rate of turnover of the head." A VERY important concept for distance casting.
3.) The idea is that when we shoot out line into the running (shooting) line, that we gain distance by causing a delay in the completion of turnover of the head loop. That means we literally slow down that turnover, even though we don't slow down its forward motion. Now, the longer it takes for that loop to unroll, the longer the cast since forward motion of that loop only occurs as it unrolls.......as long as the whole thing doesn't collapse to the water.
4.) We do this by allowing the shooting line to escape the line hand and rod tip. This results in less resistance than we had when casting with the line stationary at the rod tip.
5.) That remaining resistance is the inertia produced by friction at several points including the line hand, the guides, the tip top, any line contact with the rod, as well as the "skin friction" of the line itself against the atmosphere. To this is added the changing resistance produced by the weight of line between the stripper guide and the ground or boat deck.
6.) It doesn't change the 2:1 ratio between the velocity of the fly and that of the forward motion of the loop.....but necessarily modifies both.
7.) With the greater weight and length of line out of the rod tip, gravity acts over a longer period of time to yield not only descent of the line, but to produce a slight belly or, "sag". This increases air drag and decreases the forward velocity of the loop.
8.) Now, in your, "frictionless world", these changes would be a great deal less......but we're casting in the real world. Re your 2nd paragraph, I submit the loop begins to form at RSP whether or not in a frictionless world. (Although I guess the line would be harder to handle in that world.) since this is the point at which the fly line can overtake the rod tip.
9.) Your statement, "....The rod leg speed and the fly speed would equal the loop speed." Isn't true. With the shoot, the fly is still moving a lot faster than the loop despite the fact that the rod leg is escaping the rod . That rod leg speed is a lot less than the fly speed, too.
10.) I think you may be correct about the perfect shoot...although another way of looking at that is that there occurs full dissipation of energy as the fly leader unrolls with efficient transfer of energy in, "the perfect shoot" a well as the, "Perfect cast."
To find out what really happens, we need measurements of velocity which we can record. (DesCartes once said, "To measure is to know.")
As I consulted the physics literature, I did not find that any such measurments have been done and published. That isn't surprising in view of the interdependent variables which enter the picture when shooting line.
This would make a good physics thesis at a PhD level.
Let's bounce this to Bruce Richards who just might inform us that he and Noel Perkins (Physicist) have been working on it........................(????)
Gordy
From: "North Fork Flies - Tom Cooper" <cooper@xxxxxxxxxxxxxxxxxx>
Reply-To: "North Fork Flies - Tom Cooper" <cooper@xxxxxxxxxxxxxxxxxx>
To: "Gordon Hill" <hillshead@xxxxxxx>
Subject: Shooting line and line speed
Date: Thu, 23 Feb 2006 14:52:08 -0500
Gordy,I have been trying to imagine the mechanics of shooting line and the speed of the fly. I am doing this while avoiding viewing videos of the fly cast to see if my minds eye has a firm grasp of what is happening.First, in a frictionless world, the loop is created at the Rod Straight Point as is the beginning of the shoot. Imagine marking the center - or leading edge - of the loop. In our frictionless world that mark would stay in place during the entire shoot. The rod leg speed and the fly speed would equal the loop speed. The perfect shoot would end when the amount of line from the rod tip, around the loop, and back the fly leg to the RSP would equal the amount of line held at the end of the previous cast. One-third (or the shoot distance) would be behind the RSP.NOTE: If we held 60' at the end of the last cast there would be 30' out on the rod leg to loop center, 30' out from the center of the loop back the fly leg to above the rod tip, and 30' from that point back to the fly leaving 60' on the fly leg. (At a loop speed of 80 feet per second the fly would touch down in under .375 seconds).The shoot is completed and at this point the rod leg becomes fixed, the loop unfurls, and the fly speed suddenly accelerates to 2X the loop speed. (It seems this should set off a shock wave that would appear on the rod leg just behind the loop that would travel toward the rod tip.) The mechanics of fly speed we have discussed take over at this point.Now if my thinking isn't totally flawed and we add friction and what others forces would act on the speed of the rod leg - I'm not going to try to figure that out- I think it would go something like this. Let's say the decrease in the rod leg speed, due to whatever systems work on it, is 10 feet per second. Our Rod Leg speed is 70 fps. Using the formula 2X(Loop Speed - Rod Leg Speed) + Loop Speed = Fly Speed. So for our example it would be 80 fsp - 70 fsp = 10 fsp times 2 = 20 fps. Add 20 fsp to 80 fsp and our fly speed is 100 Feet per second. I'm sure the forces acting on the shoot would be dynamic and ever slowing so let us use a rod leg speed at the end of the shoot of 10 feet per second. 80 - 10 = 70 x 2 would be 140 fps. Add the 80 fps Loop speed and we end up with a fly speed of 220 fps. Hum. That would result in a reduction of fly speed at the end of the shoot of 80 fps!I have either shown that my logic is totally WHACKED or that if Rod Leg or Shoot speed falls below one-half of the loop speed a bad cast will result. Our formula would show that 80 fps - 40 fps = 40 fps. 40 fps X 2 = 80 fps then add Loop Speed and we end up with 160 fps Fly Speed.Gordy, I'm not sure if there are factors involved that limit the speed of the fly to 2X the loop speed or not. I'm also not certain if my logic is totally flawed or just a little off. This was merely an attempt to see if my native logic put me in the ball park of understanding the mechanics of shooting line and fly speed.Thanks and be gentle.Tom Cooper