Sunday, January 23, 2011

The Physics of Whip Cracking

I tried my hands on whip cracking in an Aussie sheep farm tour late last year.  Interestingly, whip cracking in Australia has gained enough prominence to be a competitive sport! I must admit that the whip is not my forte, as I could only get it cracking 10% of the time (when I am not hitting myself with it).  Staring at defeat, I decided to learn the science behind this in hope of conquering it some day.

It turns out a whip cracks because a sonic boom is created at its tip or topper. A sonic boom is created when an object travels at supersonic speeds such that its surrounding waves of air pressure are compressed into a single shock wave at the speed of sound. The logical question then: what is travelling so fast in the whip? Well, it's the tip.

Why? Conservation of momentum! Let's picture this: when you apply a force to a whip, the initial loop motion and wave are applied to the thong, which has a much larger mass than the whip's tip. It is worth noting that momentum is the product of the mass and speed of the moving object. The smaller the mass, the higher the speed, thus the lighter tip ends up moving extremely fast. In fact, a tip can reach Mach 2 and 30 times its initial speed when whipped properly. 

Now that I know the science behind whips, hopefully I will see improvement next time I get my hands on one.


-PTS

Sunday, January 16, 2011

Prime Numbers and Encryption

Want to make $250 000?  Find a big prime number, a really big one. It turns out there are organizations  ready to dough out good cash for a really large prime number.  This is because primes are used in  RSA cryptography.

RSA Algorithm
Let's look at the algorithm:

1.      Multiply two large prime numbers p and q to get the product N

2.      Find two numbers e and d, such that ed = 1mod((p-1)(q-1)), where e and N are relatively prime meaning they do not share any prime factors.
3.      Let's call M the original message and C the ciphered message: 

a.      To encrypt: C = Memod(N)

b.      To decipher: M = Cdmod(N) 


In essence, using the public key (N,e) will transform the original message M to the ciphered message C. On the contrary, applying the private key d on the ciphered message will result in the original message M

Security of Encryption
The beauty of RSA is your public key can be published for anyone to encrypt a message that only you can decipher. This is because only you possess the private key, and it is extremely difficult for others to deduce this from the public key unless they can easily factor into and q. Prime factorization of a large number is a tedious process which cannot be automated. One must therefore resort to brute force when attempting to crack the code.

You may have guessed that the size of prime numbers used dictate the strength of the encryption. A message encrypted with 5-digit prime numbers (40-bit encryption) yields about 1.1 trillion possible results.  However using 16-digit numbers (128-bit encryption) generates 340,282,366,920,938,463,463,374,607,431,768,211,456 possible combinations.  Based on today’s computing power, a 40-bit, 56-bit, 64-bit, and 128-bit encryption can be respectively broken in 1 second, 19 hours, 7 months and 11,000 quadrillion years. This is why 128-bit encryption is the standard used world wide to protect financial transactions and sensitive data. Don't get too complacent however, as it has been predicted that 128-bit encryption will be breakable in about 100 years.

How Many Prime Numbers Are Out There?
It is no secret that there are an infinite number of primes. Here is a proof from the famous Euclid:

1.      Assume there is a largest prime number, p

2.      Create a new number q, equal to the product of all primes between 2 and p, plus 1.

3.      Our new number q has no factors in the original set of primes (between 2 and p), because dividing by any of them would produce a remainder of 1.

4.      From point (3) we conclude that q is either itself prime, or is composed of prime factors, all of which are larger than p

5.      Point (4) falsifies point (1).


We can take comfort that the next prime number is out there. Unfortunately, it will probably take you a long time to find it.  The current largest prime243112609-1, was discovered in 2008. So just how big is this Mersenne Prime? It has 13 million digits, and will take me 8 weeks to write out the number.


-PTS

Sunday, January 9, 2011

The Colours of Water - From The Caribbean to the South Pacific

Last year, I took a number of vacations which involved cruising, diving, snorkelling and walking on glaciers. Thinking back on those trips, I could not help but notice that water and ice exhibited different colours in different settings.  I decided to dive deeper into the colours of water. Finally, here are my findings.

Water’s True Colour
The colour of an object mainly depends on the colour of light emitted from it. In the case of water its colour is also affected by factors like light source, absorption, scattering, and suspended materials. Absorption by water is stronger for red light, but weaker for blue light. Water is therefore, intrinsically blue. However, this effect is only apparent when the water is reasonably deep. This is why a glass of water appears colourless whereas a big aquarium looks bluish through the thickness of water.

Why is the Sea Blue?
I did two cruises last year, one in the Caribbean and the other in the South Pacific. Owing to the depth of the water, these seas and oceans looked navy blue from the cruise ships! Some would also argue that the sea appears blue under a clear sky because the blue light from the sky is reflected by the water surface.

Underwater Colour
In my diving and snorkeling excursions off the cruise ships, the underwater world was often revealed behind a tint of blue. As it turns out, absorption is the most important factor affecting the colour seen under the water. Red light becomes weaker with depth because it is most easily absorbed. Beneath the surface, water also scatters blue light. Thus we see the greenish-blue effect depending on the depth. As we go deeper, we will see more blue as the greenish colours from the penetrating light rays are absorbed. The overall light intensity also decreases and there comes a depth when it is pitch black. When taking pictures under water, be sure to use a flash so that the subject’s true colours can be captured.

Waters with Special Colours
The colour of water is also affected by what it contains. Suspended materials affect the colour of water. Sea water will appear green when it contains large amounts of green algae. Muddy water has a brownish yellow tinge.

When I was in New Zealand’s South Island, I found that most of the lakes are aquamarine rather than blue. These lakes were largely formed due to glacial actions millions of years ago. The fine powders produced when glaciers grind over rocks during descent were so fine that they were suspended in these waters. These powders are great at scattering light and sending it back to the surface, making the water as a whole brighter. Furthermore, as these particles are everywhere in the water, it takes a shorter distance for the light to be scattered and leave the water. Absorption of red light is thus less, shifting the original blue colour towards green.

Glacial Colours
Thin layers of ice and snow reflect light quite thoroughly, hence often appearing white. When walking on Franz Josef Glacier in New Zealand, I noticed that the ice was blue. Like water, light is scattered and absorbed in thick ice. The longer the path light follows to reach the surface, the bluer the tint.

The purity and age of the ice also play an important factor in the colour of glacial ice. As the ice ages it is compressed, melted and frozen again eliminating air bubbles that scatter the light rays and bounce them back out in the same white colour they entered. Once the ice becomes pure the light waves are much more likely to be absorbed, hence, promoting the deep blue shade.


-PTS

Sunday, January 2, 2011

2011 New Year Resolutions

Happy New Year!  I fell prey once again to the 9 - 5 in December. Combined with the year end holiday season,  this blog saw very little action last month. However, my resolution this year is to post one article each week to make this blog an interesting read for all of you out there! I know they say that resolutions are made to be broken, but I have a great feeling about this one. See you all next week!


-PTS