I have a "correction" to the math that I posted below. Dave Kruger (bless his heart) decided to take me up on my request that someone check my math. He could not get my result, for the very good reason that there is an "inconsistency" between my description of the problem and my result. (How's that for spin? My answer was correct, but only if I redefine the problem a bit. :-)) I stated before that my result was for the case where there is no shoreline eddy protecting you from the current when paddling along the bank. But in trying to determine why Dave got a different answer, I discovered that my earlier formula is for the case where there *is* protection along shore from the current, allowing you to paddle forward along the bank at your paddling speed of "p" (as opposed to p-c, where c is the speed of the current in the channel). So here is a simple, new-and-improved summary of the two cases for st/ft, the ratio of the straight-angle time (including the upstream paddle) to the ferry-angle time: CASE 1 (new-and-improved description): This applies when there is a constant paddling speed, a constant current speed away from the bank, and a protective "eddy" (with zero current speed) next to the bank. FORMULA (as before): st/ft = (1+c/p)cos(a), where "a" is the ferry angle (in degrees), which equals the inverse sine of c/p. BREAKEVEN POINT for ferrying (as before): When "p" (your paddling speed) exceeds "c" (the current speed) by 19%, st/ft=1. If p exceeds c by more than 19 percent, ferrying will be faster (for a given paddling speed). CASE 2: This applies when there is a constant paddling speed, a constant current speed away from the bank, and NO protective eddy (current speed equals c along the bank as well as in the channel). FORMULA (new and improved): st/ft = {1 + 1/[(p/c)-1]}cos(a), where a is as defined above. BREAKEVEN POINT for ferrying: In this case there doesn't seem to be one! It is always faster to ferry across. (I will have to double check this.) I think that the above is correct, but as always it needs to be verified by someone else. Obviously there are many other cases as well. An interesting case might be to determine at what ratio of along-shore current to primary current there exists a potential savings from avoiding a ferry. This would of course depend on the ratio c/p as well. Dan Hagen Dan Hagen wrote: > > > I have worked this out for a current (using some simplifying > assumptions), but not for a side wind. The latter is more complicated, > due to the side slipping. (The current moves you sideways with the > water, but not sideways relative to the water. The wind blows you > sideways relative to the water, in addition to the former.) In the > simpler case of a current, there are some circumstances where you are > better off if you do not ferry across, but instead paddle with your boat > pointed straight across and then paddle forward along the bank into the > current to recover from your sideways drift. When would this be true? > The following draws from my earlier posting on this subject: > > For the case of a current with uniform strength, my very rough model > suggests that the break-even point is for a ferry angle of about 57 > degrees, which occurs when the ratio of paddling speed to current speed > is approximately 1.19. In other words, if you can paddle more than 19 > percent faster than the current, your ferry angle will be less than 57 > degrees, and it will take less time to ferry across than to paddle at a > straight-across heading (with zero ferry angle, followed by a paddle up > current). On the other hand, if your paddling speed is less than 19 > percent faster than the current, your ferry angle would have to exceed > 57 percent, in which case it will take you less time if you paddle a > straight-across heading (zero angle), even though you have to paddle up > current once you reach the other side. > > Now for the math! Using some fairly straightforward trigonometry, it can > be shown (if I haven't made an error) that the ratio of the > straight-angle time (including the up-current paddle) to the ferry-angle > time, st/ft, equals (1+c/p)cos(a), where "c" is the speed of the > current, "p" is the paddling speed, and "a" is the ferry angle (in > degrees) which itself is a function of c/p. The necessary ferry angle > "a" equals the inverse sine of c/p. In other words, the time ratio > st/ft is a function only of c/p. This ratio equals one when c/p is > approximately equal to .83867, which corresponds to a ferry angle of 57 > degrees. If the ratio c/p exceeds .83867 (i.e., if p/c is less than > 1.1924), then st/ft is less than one and ferrying will actually take > more time than paddling at a straight-across heading, even though you > have to paddle up current. (I have checked this against a few > simulations, and this seems to work, but as always someone should check > the math...) > > Next time you have to cross a current, ask yourself if you can paddle > more than 19 percent faster than the current. If so, then you MIGHT save > time by ferrying. I say that you might save > time, because the above analysis assumes a current of constant > strength. But I have never seen such a current. It is typically slower > near shore, even in the absence of eddies. And with eddies it is, of > course, a whole new ballgame. So actually, it is quite a bit more > complicated than in the simple model above (it always is). Add your own > fudge factor to the 19 percent rule. But perhaps the above "rule" > provides a crude staring point. I would certainly welcome a more complex > model, if someone knows of one or has the desire to develop one. > > Again, the case of a side wind would be more complicated. But given that > ferrying is not always faster for a side current, the same may be true > in the case of a side wind--but the parameters would certainly change. > > Dan Hagen > > *************************************************************************** > PaddleWise Paddling Mailing List > Submissions: paddlewise_at_lists.intelenet.net > Subscriptions: paddlewise-request_at_lists.intelenet.net > Website: http://www.paddlewise.net/ > *************************************************************************** *************************************************************************** PaddleWise Paddling Mailing List Submissions: paddlewise_at_lists.intelenet.net Subscriptions: paddlewise-request_at_lists.intelenet.net Website: http://www.paddlewise.net/ ***************************************************************************
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