There has been some talk about using excel etc to compute vmg and vmc, and I posted a comment on Schakel’s blog regarding this method of manually finding the optimum VMG angle alot more quickly than ‘randomly’ trying angles – although of course being intelligent creatures we wouldnt exactly choose random angles, but this method gets you very close in only a few calculations. So I thought I would post it here to, and expand it a little.
To find the twa for maximum vmg, you can employ a ‘binary search’ type algorithm/method. For those unfamiliar, you basically narrow in on the best angle from each side, splitting the angles as you go. For example you might start by computing vmg at twa = 40 and 50, now split them to get 45 and you note that 45 is highest, followed by 50 then 40. So now you choose 47.5 (right in the middle of your 2 best angles – the average). Say you note this is greater than vmg for 50 but still less than 45, so now split 45 and 47.5 to get 46.25. You find that this value is greater than the one for 45 so now split 46.25 and 47.5, about 46.8… repeat until you are no longer computing larger vmg values. In general just take the average of your 2 best vmg angles so far, and recompute – then repeat including the new value. For most boats starting angles of 40 and 50 are good, but some polars have max upwind vmg < 40, so perhaps starting at 35 is a good idea. Wider start angles dont add much time, as they only create a few extra steps. In the example we got to 46.8 in only 4 iterations (not including the original calculations for 40 and 50) – 45, 47.5, 46.25, 46.8 – notice how the series is quickly converging? the first ‘gap’ was 5 degrees, then 2.5, 1.25 and so on – after 8 iterations you are within 0.02 of the optimum vmg TWA – exponentially getting closer each time, and probably more accurate than the interpolation used to compute BS – meaning about 6 iterations is all you will need to do – approx 5 minutes with a handheld calculator!
Also, if you use the builtin Windows Calculator (or a handheld – and in windows Calculator you need to be in Scientific Mode – under the view menu) it will default to being in ‘degrees mode’, so you can save the step of converting the angle to radians and just enter “45 COS * BS =” to get the vmg for twa=45.
Of course, exactly the same method can be used to compute the optimum VMC angle, but using the formula BS * COS(WP_CC – CC) instead, where CC is the compass course under consideration, WP_CC is the compass course to the waypoint your are trying to maximise VMC toward, and BS is the boatspeed you would be doing on the twa corresponding to CC. But remember: you need to compute the new BS for each CC angle you try – Schakel’s DC Calculator tool is very useful for this.
Also, it doesnt matter which way around you take the difference between a possible CC and the WP_CC, because: CC – WP_CC = -(WP_CC – CC) and COS(A) = COS(-A). The method IS a little different though.
I will make another post detailing how to use this method for computing VMC quickly. But basically you apply a similar strategy in varying your choices of CC.