Pilot's Rule of 60Date: 06/07/2002 at 17:31:53 From: Charlie Subject: trig/piloting: the rule of 60 There is a famous piloting rule, called the "rule of 60," that applies when you know your position and you have an obstacle ahead. Point your craft at the obstacle and think this, "60 times the desired distance off over the distance ahead equals my course change in degrees if I want to avoid the obstacle." Is there a formula describing "the opposite" of this process? You know your position, distance and degrees away from the obstacle, what will be the "closest point of approach"? Algebraic reconfiguring of the "rule of 60" results in wild innacuracies and I'm just not good enough to figure it out. Charlie Date: 06/07/2002 at 23:29:50 From: Doctor Peterson Subject: Re: trig/piloting: the rule of 60 Hi, Charlie. I think the "rule of 60" is an estimate of the following problem: B + | \ d | \ | + C | / | / r| / | / | / |A / | / |/ + A Here you are at A, r units away from an obstacle at B, and you want to turn A degrees to the side so you will be d units from B at closest approach, which is when C is a right angle. The exact solution is trigonometric: sin(A) = d/r so you want the angle A = sin^-1(d/r) using the inverse sine. Since, for SMALL ANGLES, the sine of an angle is approximately equal to (but slightly smaller than) the angle expressed in radians, we can say d/r =~ A * pi/180 and, solving for A, A =~ 180/pi d/r Since 180/pi is about 57 (the number of degrees in one radian), this gives essentially your formula, A =~ 60 d/r Using 60 makes A a slight overestimate, compensating for the fact that our sine approximation is a slight underestimate. In fact, the angle will reach exactly 60 d/r when d/r is exactly 1/2, and the angle is 30 degrees. Up to that angle, this is a pretty good approximation, never off by much more than 0.5 degree, and is off in the direction you want to be off for safety. Past 30 degrees, however, it starts to get pretty bad! Now, you'd like to reverse this and find d given r and A. Again, the exact solution is d = r sin(A) If we instead solve our approximation above for d, we get d =~ rA/60 which, as before, is an appropriate estimate, underestimating your closest approach (by no more than 5%) for angles under 30 degrees. So the rule is: multiply your distance from the obstacle by your course (relative to the obstacle) in degrees, and divide by 60 to get your distance at closest approach. As long as the angle is less than 60 degrees, this will work well. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/08/2002 at 14:17:24 From: Charlie Subject: Thank you (trig/piloting: the rule of 60) Thanks ever so! I am a 36 year-old dyslexic, always terrible at math and strong on concept. You explained it perfectly! I will memorize it as a word problem and test it on a flight simulator. Then I'll take it sailing! My best to you who output such good into the world! |
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