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At this time of year a lot of people dream about ‘getting away from it all’. So how about REALLY getting away from it all?

REFER TO DIAGRAM

Newton’s Gravitational Law says that:
F = GMm/r²
Where F is the force of gravity, G is Newton’s Universal Gravitational Constant, m and M are the masses the objects feeling the force, and r is the shortest distance between them.
In this equation forces are related to energy change by integrating the force with respect to ‘r’ (A level) or force x distance (GCSE). The graph of the force against ‘r’ looks like this (r is labelled as distance):
Now I don’t like the look of multiplying F x r so I’m going to use the A level approach and integrate this, in other words,∫_R^∞▒〖GMm/r^2 〗 . dr, if I really aim to get away from it all (an infinitely long way).
The result of this is that the energy required is given by –GMm/r (Notice the energy is negative, in other words it costs energy to increase r). Using the limits of integration this gives GMm/R where R is the radius of the earth. If you turn this into numbers (and if your mass is 70kg) you get:
Energy required = 6.67 x 10-11x 5.98 x 1024x 70/6.37 x 106 Joules
= 4.38 x 1013 Joules, which if you could extract all of the energy from some petrol would leave you needing about 130 litres, at a cost of about £160. Looks good so far but then you’ve got to factor in survival gear (most of the journey is in a vacuum) and the fact that you’ll need food and drink for the rest of your life (you will die before you get there as you are travelling to infinity and therefore at any finite speed it will take an infinitely long time to get there!). Perhaps a week in Spain instead?

How to show that cosec 2x + cot 2x = cot x
cosec 2x + cot 2x = 1/sin 2x + 1/tan 2x =
1/sin 2x + cos 2x/sin 2x
(1+ cos 2x)/sin 2x = (1 + cos²x - sin²x)/2sinxcosx =
(1+1 -sin²x - sin²x)/2sinxcosx
(2 – 2sin²x)/2sinxcosx = (1 - sin²x)/sinxcosx =
cos²x/sinxcosx = cosx/sinx = cot x QED

A bit more on Pythagoras. Before any aspiring astrophysicists jump all over me, there’s just one tiny detail missing… what if space isn’t flat?
Say what? Well our space exists as one of three possibilities; spherical, flat or hyperbolic! One way to appreciate the difference is to go on a journey with a friend. Start off going in exactly the same direction and in flat space you will always stay the same distance apart what ever the length of the journey. OK so next time you are on the equator going for a stroll start walking north with a mate. If you keep on going in flat space you should never meet, but actually what will happen is that you will meet when you get to the north pole. So the surface of the earth isn’t flat! (I think most of us knew that already) The surface of the earth is a sphere (roughly) so spherical space is reasonably easy to understand. Getting a bit technical, in spherical space parallel lines converge. In hyperbolic space they get further apart. Simples.
And getting to the point, Pythagoras Theorem is only true for flat space.

Interesting

Experimental test of local observer independence The scientific method relies on facts, established through repeated measurements and agreed upon universally, independently of who observed them. In quantum mechanics the objectivity of observations is not so clear, most markedly exposed in Wigner’s eponymous thought experiment where two observers...

A theorem is always true….. so how did Pythagoras know this? Well first you need two squares of different sizes. Then you do this with them:
Truth be told I’m not sure that this is exactly how he did it, there are several other methods to get the same result, but this is the one I like best.

A side of the large square is a+b, so the area is (a+b)²
Expand the bracket and you get a²+2ab+b².
Now if you find the areas of all of the triangles formed at the corners and add them together you get 4x(½ab)=2ab
Next take the triangles away from the large square and you have an area of a²+2ab+b²-2ab = a²+b².
But hey that leaves the grey square which has an area of c²
So a²+b²=c² and the rest is history!