Dork Scratchings

How fast would you have to fire a cannonball for it to never hit the ground? Newton's very first ideas about gravitational orbits are said to have come about from a thought experiment. A cannonball was known to lose 5 metres of altitude a second after being fired horizontally, but the Earth - being round - curves away from the cannonball as it flies forward. So it occurred to Newton to ask: How fast would the cannonball have to be fired for the curvature to completely compensate for the vertical loss? If a cannonball was fired at this speed it would never lose any altitude, and end up orbiting the Earth. The diagram above shows that the answer can be found using simply trigonometry and comes to $$ \begin{align} v &= \sqrt{gr} \\ &= \sqrt{9.81 ms^{-2} \times 6.371\times 10^6 m} \\ &= 7868\space ms^{-1} \\ &= 17603\space mph \end{align} $$ In general, (non-relativistic) orbits are elliptical The next stage was to look at more general orbit...

Is Our Universe Finite? A while ago I drew the pictures above to try to understand current ideas about the size of the universe. The diagrams are based on some pictures I saw in the book " Our Mathematical Universe ". The diagrams show two dimensional slices of four dimensional spacetime. The blue stuff is "inflationary material" which expands at an enormous rate. The current theory of inflation states that universes like ours form as bubbles in the inflationary material as some of the inflationary material changes phase and "evaporates" out as non-inflationary material. An important point is that the sides of this bubble are moving away from each other way too fast for anything - even light - to travel from one side to the other.  The 1st diagram illustrates the point that in this model there is room for more, far more, than one universe. The yellow region in the 2nd and 3rd diagrams is what is known as a light cone. The point in the middle...

Any orbit can be represented given enough epicycles. The Geocentric Model The geocentric model was wonderful for human self aggrandizement.  We belonged at the centre of the universe and everything revolved around us.  That early humans believed this isn't that surprising.  After all the Sun, moon, and stars do seem to move in circles around us (albeit different ones).  What's more interesting is how we attempted to cling on to this theory in the light of a) conflicting evidence, and b) a far better explanation. David Deutsch in The Fabric of Reality puts forward the view that the point of a scientific theory is to explain .  The more a theory explains - whilst still remaining consistent with observable facts - the better it is.  This is a philosophical justification for Occam's Razor.  Why is an explanation with fewer postulates better than one with more postulates? because it leaves less unexplained! The earliest theory for why the plane...

Queqiao and Chang'e-4 Exciting news from the BBC Website: China Moon mission lands Chang'e-4 spacecraft on far side .  The page includes a video with sinister background music as if to suggest they're the baddies (a la Drax in Moonraker).  However, the part that really intrigued me was the mention of the L2 Lagrange point - the first I've ever seen in a news story! As I described in my post Lagrange Points   there are 5 locations in the Earth-moon-Sun plane in which - in the rotating frame of reference and taking centrifugal forces into account - there is no overall force and an object can be parked indefinitely.  One of them is just beyond the moon and is called L2 . Now the problem with landing a probe on the far side of the moon is that you can't talk directly to it: there's a big rock in the way!  So, according to the BBC article the Chinese Space Agency has parked a satellite Queqiao at L2 to relay messages.  This left me a bit confused...

If the sky were covered in moons it'd be almost a bright as day! Walking to the pub through the Yorkshire Dales on a particularly brilliant moonlit evening, when everything was clearly visible, I wondered just how much less light there was than during daytime?  It turns out to be quite easy to estimate an upper bound - all you need to do is measure the angle subtended by the moon! Let $A_e$ and $A_m$ be the cross sectional areas of the Earth and moon, $d$ be the distance to the moon, and $r$ be the distance to the Sun.  Now, suppose the Sun releases some energy $E_s$ then the amounts $E_{se}$ and $E_{sm}$ which land on the Earth and the moon are given by: $$ \begin{align} E_{se} &= \frac{A_e E_s}{4\pi r^2}\\ E_{sm} &\approx \frac{A_m E_s}{4\pi r^2} \end{align} $$ On a full moon, let's assume for the sake of calculating an upper bound that the moon reflects all of $E_{sm}$ equally in all hemispheric directions.  Then the energy reflected to the Earth ...

Imagine you were a hamster in a hamster ball living on a hilly surface.  Base camp is surrounded on all sides by high summits, but you have a powerful catapult there that can fire you to the top of any of them.  Once fired you can influence your trajectory, but it's hard work and you don't have much energy in your little legs.  Suppose you know where you want to end up beyond the hills.  What's the best strategy for getting there?

COBE CMB fluctuations. Original Source: NASA All tortoiseshell cats are female.  Males can be black, or ginger, but never tortoiseshell.  The reason for this is that the mechanism by which tortoiseshell cats get the patterns on their coats depends on having two X chromosomes.  This is all described beautifully in Chapter 7 of "Junk DNA", by Nessa Carey . Females have twice as many X chromosomes as males, which on the face of it should result in 100% more expression for the genes on that chromosome.  This should lead to much greater differences between males and females than we actually see.  To put this in perspective, Down's syndrome is caused by individuals having 3 copies of chromosome 21 instead of 2.  But this is a far smaller chromosome than X and the difference is only 50%, rather than 100%.  (The fact that chromosome 21 is so small is the reason Down's syndrome is more common than syndromes in which there are too many copies of more impo...

A review of Our Mathematical Universe (Max Tegmark) Our Mathematical Universe was my summer holiday reading this year.  But it turned out to be much more than just something to keep me occupied while lounging on the beach.  This book has changed my conception of reality.  The themes in the book are similar in nature to those in The Fabric of Reality by David Deutch - another one of my favourite books.  However, instead of restricting the argument to the parallel universes predicted by Everett's Universal Wavefunction, Tegmark takes us on a tour of four levels of multiverse.  More than this, he provides an overarching theoretical framework for understanding them based on what he calls "the A-word" (because using its full name is guaranteed to get your paper rejected). The argument goes like this.  Whenever we find Nature appears finely tuned to make self-aware life possible, then there are 3 possible explanations It's a fluke! It's design! (by a...