Transcript the Force of Math

May the Force of Math
be With You!
Striking out ignorance
Star Wars - Fantasy or Reality?
• In some ways the struggle of good an evil
is a constant battle!
• Even though fantasy, a lot of fun fantasy
math can be derived
How much energy does a light saber
hold?
Pure Energy
A light Saber can cut through metal, sever a limb, fend of
bullets. How much energy must it contain?
Lets look at cutting through metal:
Suppose a light saber makes a 2 meter cut through a solid
metal door that is 20 cm thick.
The light saber is about 7 cm thick in diameter:
Great for a pre-calculus demonstrating
the use of formulas:
Thermal Energy = mass*Specific heat of
element* change in temperature
Q= m C (T2 – T1)
Also,
Q = m Lf
Lf is called the latent heat of fusion which is the
energy required to melt a kg of the material.
We will need both of these formulas…
First we have to know what type of metal. Let’s assume the
Empire uses titanium doors. The volume of door that is cut:
= 2 x 0.07 x 0.2 = 0.028 m3
Titanium has a density of 4540 kg/m3, which gives us a mass of
127.12 kg.
Assuming an initial temperature of 250C or 298K, the energy to
heat the door to its melting point is Q= m C (T2 – T1) = 127 x
0.54 x (1941-298) = 112,677 Kilojoules!
Now, we need to find the energy to convert the titanium from solid to
liquid as it is being cut. We will use the second equation:
Q = m Lf
Q = 127 kg x 302 KJ/ kg = 38354 KJ
Add them together to get: 151031 Kilo Joules
To compare: The amount of energy released by the atomic bomb dropped
on Hiroshima was about 50.2 billion KJ.
How many times more is this?
Power = Energy/Time
Suppose it took 9 sec to cut through the door, how much power
is this in watts? One watt = One Joule/second.
16781 Kilo Watts
SPACE TRAVEL
Form Coruscant to Tatooine
https://www.youtube.com/watch?v=bosSsgzgenA
Solo claims his ship travelled the Kessel run in less
than 12 parsec. What is wrong with his statement?
Parsec is a measure of distance,
not time!
• On the map it says 1500 parcec = 4893 light
years. Is this true?
• And each square is approximately 1500
parsec.
The word Parsec comes from parallax and
arcsecond.
Parsec: The distance at which one AU subtends an
angle of one arcsecond.
(in general, the subtended angle is called the parallax.)
Calculate the the equivalent miles, A.U., meters and light years for 1 parsec:
1. Arclength = angle (in radians) x radius (assume an arc between the Earth and Sun).
2. Using a right triangle trigonometry.
So let’s do some math. Lets find the equivalent of a Parsec in km
and then light years using arclength:
One A.U is approximately 93,000,000 miles or ? Km.
150,000,000 km
Arclength = radius of the arc x angle subtended (in radians)
We are looking for the length of the radius:
150,000,000 = one parsec x (1/3600) x (pi/180)
So one parsec = 150,000,000 x 3600 x 180 /pi =
3.1 x 1013 kilometers or about 31 trillion km!
Let’s do the same calculation using right triangle trigonometry:
Sin (1/2 arcsecond) = 75,000,000/One parsec
One Parsec = 75,000,000 / sin(0.5 arcsecond)
= 3.1 x 1013 km
Light travels at ? Km per second.
About 300,000 km/second
A light year is the distance light travels in one earth year. So one
light year = ? km
9.5 x 1024 km
And so one parsec = (3.1/9.5) x 10 = 3.3 light years approximately.
Is the scale on the map correct?
Solo says his ship can travel this run in 12 parsec, but this is a measure of
distance which is equivalent to 39 light years.
But there is no indication as to how long it took to do this run
Now assuming the distance is 12 Parsec and travelling at the
speed of light, how long would this ship take to do this run?
39 years! (I don’t think that’s what George Lucas had in mind:)
Distance from
Corusant to
Tatooine is
about 9
squares (the
Orange Route).
Ignoring the
fact that they
are diagonals
each square is
1500 parsec or
4893 Light
Years,
travelling from
Tatooine to
Corusant
would take:
4893 x 9 =
44,037 years
traveling at the
speed of light!
To travel
that
distance in
two days
we would
have to
travel how
many times
faster than
the speed
of light?
365/2 * 44037 =
8,036,752.5
times the speed
of light!!
Introduce the
idea of
Hyperdrive
(or Warp)
Space, the final frontier…. Is interstellar travel at the
speed of light or faster really possible?
m approaches infinity as well!!
The Power of the Dark Side
“The Death Star”!
Or Star Killer Base, that harnesses the sun to destroy planets…
How much energy does it really take
to blow apart a planet?
In one of the episodes, the death star destroys the planet Alderan. How
much energy would it take to destroy a planet of this size? Lets take a
planet we know about: Earth.
We will postulate that to blow apart a planet, you need to pull apart
the bonds that bind the molecules of the earth together. We can think
of these as gravitational bonds.
There are many ways to approach this, in fact, the Department of
Physics and Astronomy at University of Leicester published a paper in
2011 on the energy required to blow up planet earth:
Using Newton’s law of gravitation F = GmM/r2 and integrating it
through each layer of a planet they postulated that the energy
required to blow up a planet is given by:
E = (3/5)GM2/R
Where G is the universal gravitational constant 6.6743 x 10-11 m3/(kg
sec2 ). M is the Mass of the planet and r is the radius of the planet.
We will make a gross generalization using the mean density of earth
as: 5.514 g/cm3 . How much is this is kg per cubic meter?
5514 kg/m3
Using a radius of 6370 km we get a volume of
1.08 x 1012 cubic km
Mass = volume x the density =
5.97 x 1024 kg as the mass of Earth.
(Don’t forget to convert cubic km to cubic m)
Hence the total energy to blow up the earth is approximately
0.6 * (6.6743 x 10-11 m3/(kg sec2 )) x (5.97 x 1024 kg)2/ 6,370,000 m =
2.24 x 1032 Joules !!
Another approach is to find the force required to split the earth into two
hemispheres. Calculus can be used to find the center of mass of each hemisphere.
Using Newton’s law of gravitation F = G mM/r2 we can the force to split apart the
hemispheres, where r is the distance between the center of mass of each
hemisphere. (Happens to be 0.75 times the radius of the planet)
This would be the force they exert on each other. Thus, to split them apart you
need an energy of this force (or more) times the distance between them (lets use
0.75 R (radius of planet) ad you get an energy of about:
1.25 x 1032 Joules !
In either case, it’s a heck of a lot of energy and this is just split them apart – not to
send them into space with a huge amount force as you see in the movie!!
So how can we relate to this energy?
The sun emits 3.85 x 1026 Watts.
One Watt is a Joule per Second.
Thus it would take 2.24 x 1032 Joules / 3.85 x 1026 Watts
= 581,818 second to harness this amount of energy from the sun!
That’s approximately 6.76 days of harnessing the sun’s energy!!
In the case of Stare Base Killer, the sun would be essentially be
depleted of energy for 6.76 days!
And that’s to barely break the planet apart. In the movie parts of
the planet are catapulted through space. They would need a force
approximately 1000 times greater to accomplish this. And, it would
take 250 years of sunlight.
But who needs a death star to destroy our blue planet?
We seem to be doing a fine job
ourselves!
Yoda and conservation of energy…
Why does Yoda spins around so much when he is a fight?
Dookus Saber is longer and
more powerful. Yoda’s is
shorter. When equating the
energy you have to aslo
account for the radius of swing
for each Saber and the kinetic
energy generated by yoda
flipping around several
times
The idea is
conservation of energy:
In order for Yoda
smaller Light Saber to
counter act the Larger
Light Saber of Count
Dooku, Yoda has build
up its kinetic energy to
be the same as
Dooku’s.
But I’ll let you do that as I have run
out of energy!