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Sunday, December 8, 2013

On the Wavelength of Santa's Sled (with a nod to FA Wilson)

About a year ago, David, M0VTG, was reading "From Atoms to Amperes" by FA Wilson.   His thoughts turned to a seasonal application of Wilson's lessons....  Thanks David! 

WAVELENGTH OF SANTA AND HIS SLEIGH

We're going to need to know the following:

Total mass of the sleigh, reindeer, Santa and his presents, and the speed at which the sleigh travels. Assuming that the sleigh travels relatively close to the surface of the earth, the mass can be regarded as the same as the weight. Now assume each reindeer weighs 100 kg. Traditionally there are nine of them so total mass of the reindeer is 900kg. Say the sleigh also weighs 500kg and Santa and his presents weigh 400kg; then the total mass is 1800kg. To get round all the children who believe in him in one night would mean him travelling at a speed of say 1000kph (277.8 m/s). (Note that scientific notation for numbers used is that used on some calculators)

So mass (m) is 1800kg
velocity (v) is 1000kph (notice the not so subtle change from a scalar quantity to a vector quantity).

Max Plank came up with a formula: E = hf relates Energy to frequency using h (Plank's constant = 6.626 E-34 Joule seconds).

Albert Einstien (as everyone knows) says that E = mc2 (where c is the velocity of light).

So it follows that hf = mc2 and therefore, m = hf/c2.

The momentum of a photon (p) is defined as mass x velocity or p = mv and as every radio amateur knows, wavelength (λ) is the velocity of light divided by frequency or λ = c/f (or f/c = 1/ λ).

So mv = hf/c2 x c (velocity of a photon is c don't forget); so p = hf/c

Since f/c =1/ λ, then p = h/λ i.e. λ =h/p

Also since p = mv then λ = h/mv (This expression is known as the de Broglie wavelength - Louis Victor de Broglie)

Notice that the velocity of light does not directly feature is the expression. We can, therefore use it the calculate the wavelength of anything!

So the wavelength of Santa's sleigh is Plank's constant divided by the mass times the velocity or

λ = 6.626 E-34 J s / (1800 kg x 277.8 m/s) which equals 1.325 E-39 m - an unbelievably small number! Make up you own mass and velocity for the sleigh if like.


I defy NORAD to track such a short wavelength.


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