Intereting Posts

Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\kappa$
$A^2+B^2=AB$ and $BA-AB$ is non-singular
Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?
Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?
Geometric interpretation for sum of fourth powers
If ord$(a)=m$, ord$(b)=n$ then does there exist $c$ such that ord $(c)=lcm(m,n)$?
How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
$k$-jet transitivity of diffeomorphism group
General Continued Fractions and Irrationality
Mrówka spaces are first-countable
Is the center of the fundamental group of the double torus trivial?
Does strong law of large numbers hold for an average of this triangular array containing “almost i.i.d.” sequences?
Proof of upper-tail inequality for standard normal distribution
Why do complex functions have a finite radius of convergence?
an indefinite integral $\int \frac{dx}{\sin{x}\sqrt{\sin(2x+\alpha)}}$

Problem: A pilot wishes to fly from Bayfield to London, a distance of 85 km on a bearing of 160°. The speed of the plane in still air is 250 km/h. A 20 km/h wind is blowing on a bearing of 030°. Remembering that she must fly on a bearing of 160° relative to the ground (i.e the resultant must be on that bearing),

- find the heading she should take to reach her destination.
- how long the trip will take.

I’ve drawn the diagram for this problem numerous times and I cannot seem to figure out where to start… I know the answers to the questions because of other websites, but the answers they give don’t make sense to me as they aren’t very explanatory (it just shows the equations and how to get numbers)… I’m looking for a more thorough explanation for an answer as I’m trying to understand the problem (not just find the answer).

Please help!

- Optimisation Problem on Cone
- Why doesn't it work when I calculate the second order derivative?
- Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$
- $f(x) - f'(x) = x^3 + 3x^2 + 3x +1; f(9) =?$
- Proving that a definition of e is unique
- Uniform continuity of $x^3+ \sin x$

- Every compact metric space is image of space $2^{\mathbb{N}}$
- Finding cubic function from points?
- Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
- How is the concept of the limit the foundation of calculus?
- Holomorphic function with zero derivative is constant on an open connected set
- Energy for the 1D Heat Equation
- Proving $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}(\pi^2+12\pi \log{2}+24\log^2{2}) $
- proof of l'Hôpital's rule
- Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$
- Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$

To get the concepts going:

- Draw a line 20km long on a bearing of 30°. This is where the plane would have got to after an hour if it had been hovering stationary in the moving air.
- Pick a bearing – any bearing, and, starting at the end of the previous line, draw a line 85km long in that direction.

The end of the second line shows where the plane will be after an hour’s flight along that heading. *The “do all the wind and then do all the flying” approach does not introduce errors. To see this, draw two lines half the length, representing half an hour’s flight, then do it again, representing the second half-hour. You will end up in the same place as in the original “1-hour” exercise. If you have the patience, you can repeat the exercise with 60 minute-long segments, or 3600 second-long ones,…*

You want to be on a bearing of 160° from your starting point. So draw an infinitely long line on that bearing.

Now the heading you should actually be flying on is the heading which, after step 2, intersects that infinite line. In your place, I’d do it like this:

- Draw the 20km line on a bearing of 30°.
- Draw the infinite 160° line.
- Set your compasses or dividers to straddle a distance of 85km. Put one leg on the end of the 20km line and the other leg on the infinite 160° line. For convenience of explanation, draw a line between the legs.

The direction of the line you have just drawn is the heading to be flown.

The distance between the start of the 20km line and the end of the line you have just drawn is the ground distance flown in an hour. Having a ground speed in km/h you should easily be able to work out how the trip will take.

There are lots of other methods you might use to get the answer (for instance, fly first and get blown after), but I hope that this approach will get you to understand what is really going on.

And what happened to nautical miles, anyway?

Here’s a translation of Martin Kochanski’s excellent explanation of the concepts behind the solution into mathematical manipulations.

The solution amounts to finding the intersection between the line to the destination and a circle with radius equal to the airspeed and center that’s been offset by the wind speed and direction. A brute-force approach is to set up the equations of this circle and line, solve them simultaneously, select which of the possible two solutions is the correct one, then find the slope of the resulting line and find its arctangent (taking care to get it in the correct quadrant) to get the required heading. If there’s no solution, you can’t get there from here—the plane can’t fly fast enough to overcome the wind. This is a tedious computation. You can do something simpler by taking advantage of the geometry.

Referring to the diagram above, the line $OA$ is the “infinite line” to the destination with bearing $\beta$ and $OW$ is the wind vector with direction $\alpha$ and distance equal to the wind speed $w$. The circle is centered at $W$, while $WA$ is a radius with length equal to the airspeed $a$. The heading that we need is in the direction of $WA$, which you can see is $\beta+\phi$ by drawing a line parallel to $OA$ through $W$. By the Law of Sines, we have $${\sin(\beta-\alpha)\over WA}={\sin\phi\over WO}$$ which becomes $$\sin\phi=\frac wa\sin(\beta-\alpha)$$ after substituting and rearranging. If $\sin(\beta-\alpha)$ is negative, we’ll end up with a negative value for $\phi$, but that’s as it should be: in this case the wind is blowing to the right relative to the desired bearing, so the plane’s heading needs to be adjusted counterclockwise.

- Why is the axiom of choice separated from the other axioms?
- How to find the maximum diagonal length inside a dodecahedron?
- A question from Kunen's book: chapter VII (H9), about diamond principle
- How can I prove $dz=dx+idy$?
- Measure of the Cantor set plus the Cantor set
- A game with two dice
- Find limit $a_{n+1} = \int\limits^{a_n}_0 \bigl( 1+\frac 1 4 \cos^{2n+1}t \bigr) \, dt$
- Divide by a number without dividing.
- Can any linear transformation be represented by a matrix?
- Is it always true that $\det(A^2+B^2)\geq0$?
- Proof of 1 = 0 by Mathematical Induction on Limits?
- Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $
- Proof of the definition of cardinal exponentiation
- Values of $\gcd(a-b,\frac{a^p-b^p}{a-b} )$
- Is it impossible to recover multiplication from the division lattice categorically?