How much energy would it take for Santa Claus to travel
from home to home over the face of the Earth delivering toys?
To complete the task in a single night it would take
3.382x1020 J
or
3.186x109 times
the annual world energy consumption.
This analysis is an update of the
Classic Physics of Santa Claus
to metric units,
eliminating a factor-of-ten error,
removing erroneous religious correlations,
providing an improved power-law distance estimation,
and removing nonphysical air-resistance information.
This analysis begins with the assumption that there is a correlation
of
α = 0.8 between being a member of the
most developed countries (MDC) and having a culture which
includes Santa Claus in its belief structure.
I assume anecdotally that members of the MDC correlate
well with the belief of Santa Claus in their
culture, whereas those who are not members of the MDC do not.
The benefit of working with members of the MDC is that the
analysis can be grounded on existing census and
energy consumption data.
There are
Nmdc = 1.29
billion people in the MDC of which
β = 0.25
or 240 million are children.
There are
f = 0.51
children per home which gives
n = 122.4 million
homes that must be visited by Santa.
Nmdc = 1.2x109 people
α = 0.80 believers/person
β = 0.25 children/person
k = α β Nmdc = 2.40x108 children
f = 0.51 children/home
n = f k = 1.224x108 homes
Taking into account the rotation of the Earth,
using an eight-hour night,
and assuming Santa travels from East to West,
he will have 32 hours (115,200 seconds) to complete his work.
That works out to 942 microseconds per home.
Assuming Santa spends half of his time traveling
(
ft = 0.5)
he will have only
471 microseconds to travel from house to house.
to = 32*60*60 = 1.152x105 s
th = to/n = 9.412x10-4 s/home
ft = 0.5
tt = ft th = 4.706x10-4 s/home
The total land area of the Earth is
Ae = 1.5x1014 m2.
Populations tend to cluster according to a power law.
Stated simply, 90% of the population is concentrated in 10% of the area,
while the other 10% of is spread out over the remaining 90% of the area.
Assuming that the members of the MDC occupy the entire globe in some part,
the total area covered by Santa can be approximated according to the
power law by using one tenth of the total land area
(
Ar = Ae/10 = 1.5x1013 m2).
Divided equally, each home will occupy
Ah = Ar/n = 1.225x105 m2
of land.
Using a flat-Earth approximation,
each home will be separated by a distance of
dh = 2 rh = 2.793x102 m.
Assuming there is a solution to the traveling salesman
problem, whose solution is the sum of minimum distances,
the total distance Santa must travel is
dt = n dh = 3.419x1010 m.
This distance is X times farther than Y.
Ae = 1.5x1014 m2
Ar = Ae/10 = 1.5x1013 m2
Ah = Ar/n = 1.225x105 m2/home
rh = √(Ah/(2π)) = 1.397x102 m
dh = 2 rh = 2.793x102 m
dt = n dh = 3.419x1010 m
Now that the time of flight and total distance has been solved
for the average velocity of
<v> = 5.935x105 m/s can be calculated.
Note that this is 1746 times the speed of sound.
<v> = dh/tt = 5.935x105 m/s
vs = 340 m/s
rs = <v>/vs = mach 1746
The ultimate goal of this trip is to deliver gifts.
Assuming that each child receives a single 1-kg gift
The mass of a fully-loaded sleigh is
mi = k mt = 2.4x108 kg.
The average sled mass is
<m> = mi/2 = 1.2x108 kg.
mt = 1 kg
mi = k mt = 2.4x108 kg
<m> = mi/2 = 1.2x108 kg
The energy required to accelerate and decelerate
this mass between each home is calculated
by solving for the force during acceleration and deceleration
(
F = m a)
and multiplying it by the distance traveled
(
W = F d).
Assuming a generous reindeer efficiency of
ε = 0.5,
the total energy required to deliver the gifts is
Wt = 2 n/ε W = 4.140x1028 J.
Estimating the total yearly world energy consumption
as 1.5 times the energy consumption of the United States,
it can be seen that Santa would need
3.2 billion times as much energy as the entire world consumes
in a year, to deliver gifts in the alloted time.
ε = 0.5
a = 2 (dh/2)/(tt/2)2
F = m a
W = F dh/2
Wt = 2 n/ε W = 4.140x1028 J
r = 1.5
We = r 8.661x1018 = 1.299x1019
rW = Wt/We = 3.186x109
The average output power over the delivery period would be
<P> = 3.593x1023 W.
1.691x10
20 J lost to the 0.50 efficiency is radiated,
the power density of the radiation from the sled flying
at an altitude of 1000 m would be
At = 2.860x1016 W/m2.
For comparison the power density experienced by standing on the
surface of the Sun, given by
As = Ps/(4πrs2)
would be 6.294x10
7 W/m.
Thus Santa's sleigh at an altitude of 1000 m would deliver a
power density more than 454 million times that than of standing on the sun.
<P> = Wt/to = 3.593x1023 W
rg = 1000 m
At = P/(4πrg2) = 2.860x1016 W/m
Ps = 3.826x1026 W
rs = 6.955x108 m
As = Ps/(4πrs2) = 6.294x107 W/m
rA = At/As = 4.543x108
This proves via the principles of cognitive dissonance,
that Santa therefore uses warp drive.
□ END
| Name |
santa_claus.m
|
| Version |
1.2 |
| Updated |
2004/01/01 01:26:32 |
| RCS |
santa_claus.m,v 1.2 2004/01/01 01:26:32 forman Exp forman |
| Rating |
|
| Category |
Matlab |
| Description |
The energy required for Santa Claus to deliver his gifts on Christmas.
|
| |
|
%% $Id: santa_claus.m,v 1.2 2004/01/01 01:26:32 forman Exp forman $
%%
%% Description:
%% The energy required for Santa Claus to deliver his gifts on Christmas.
%%
%% Author: Michael Forman <Michael.Forman@Colorado.EDU>
%% URL: http://www.Michael-Forman.com
%% Creation Date: Date: 2003/12/19 10:14:23 GMT
%% Last Revision: $Date: 2004/01/01 01:26:32 $
%% Revision: $Revision: 1.2 $
%%
%% Copyright (C) 1999-2004 Michael Forman. All rights reserved.
%% This program is free software; you can redistribute it
%% and/or modify it under the same terms as Perl itself.
%% Please see the Perl Artistic License.
%%
%% Category: Matlab
%% Rating: 2/5
%%
%% <meta name="title" content="The Physics of Santa Claus.">
%% <meta name="description" content="The physics of Santa Claus.">
%% <meta name="abstract" content="The physics of Santa Claus.">
%% <meta name="keywords" content="matlab script Santa Claus physics delivery">
%
mdc = 1.2e9; % 1.2 billion people in MDC
a = 0.8; % correlation between MDC membership and belief in SC
b = 0.25; % children in MDC population
k = a*b*mdc; % number of kids = correlation * percent kids * pop
f = 0.51; % kids per household
n = f*k; % number households with kids
to = 32*60*60; % travel time in seconds
th = to/n; % time per household
tt = 0.5*th; % time in transit per household
Ae = 1.5e14; % total land area of earth
Ar = Ae/10; % area where 90% of travel will be
Ah = Ar/n; % area per household
rh = sqrt(Ah/(2*pi)); % household radius (assuming local flat earth)
dh = 2*rh; % distance between homes
dt = dh*n; % total distance traveled
va = dh/tt; % average house-to-house speed
vs = 340; % speed of sound at sea level is 340 m/s
rs = va/vs; % mach number
mt = 1; % toy weight in kg
mi = mt*k; % initial toy mass
m = mi/2; % avg toy mass
e = 0.5; % efficiency
a = 2*(dh/2)/(tt/2)^2; % accelerate half way there
F = m*a; % force
W = F*dh/2; % work
W = 2*W; % work for accel and decel
W = W/e; % account for efficiency
Wt = W*n; % total energy for all households
We = 1.5*8.661e18; % total world E consumption = 1.5 * U.S. consumption
rW = Wt/We; % ratio of Santa to world E consumption
P = Wt/to; % average power output over total time of flight
rg = 1000; % flight altitude
As = P/(4*pi*rg^2) % power density on sphere (W/m)
Pu = 3.826e26; % power output of sun
ru = 6.955e8; % radius of sun
Au = Pu/(4*pi*ru^2); % power density on surface of sun (W/m)
rA = As/Au % ratio of santa power density and sun surface p den
