THE PHYSICS OF 'FROZEN'

You've no doubt noticed that Elsa's magic is incredibly
powerful.  In 'Frozen,' she raises her towering ice
palace in just a few minutes ( and she freezes all of
Arendelle, unintentionally ) .  In 'Frozen 2,' she can
instantly freeze a wave in the sea.  Maybe we can better
appreciate how impressive her magic is by thinking about
the physics of what she did.

( Granted, it may not make a lot of sense to analyze her
magic in terms of physics, but it's still a fun exercise
to play around with the numbers. )


===================
   Assumptions :
===================

In part, these are to simplify the calculations :

1. Assume Elsa froze 1,000 cubic feet of water
   ( the waves appeared to be much larger, but
   we're more interested in just getting a
   ballpark number here ).

2. Assume the water temperature was 20 degrees
   Celsius, or 68 degrees Fahrenheit.

3. Assume we can overlook the fact that the sea
   water isn't pure water, but has plenty of
   salts dissolved in it, which lowers the
   temperature at which water freezes.


====================
   Calculations :
====================

First, the mass of the water is :

   1,000 cubic feet of water

   = 62,400 pounds

   = 31.2 tons  ( water is *heavy* )

   = 28,305 kilograms

   = 2.83 x 10^7 grams


In terms of thermodynamics ( rather than magic ),
the heat removed from each gram of water is :

   20 calories, to cool it from 20 deg. C to 0 deg. C

      plus

   80 calories, to freeze water at 0 deg. C to ice
      at 0 deg. C  ( that's the heat of fusion )

   = 100 cal. / gm.


Therefore the total heat or energy involved is :

   2.83 x 10^7 grams  x  100 cal. / gm.

   = 2.83 x 10^9 calories

   = 2.83 x 10^6 kilocalories ( kcal )

   = 1.186 x 10^10 joules ( J )

   = 1.122 x 10^7 BTU

   ( 1 cal. = 4.191 J ; 1 BTU = 252.16 cal. )


( TBD : a frame of reference for the result.
It's a big number, but what does it mean? )


====================
   Extra credit :
====================

( These are left as exercises for the reader, as they
say in textbooks. )


[ 1 ]  At the end of 'Frozen 2,' ( SPOILER ALERT ) Elsa
throws up a huge wall of ice, stopping the flood bearing
down on Arendelle.  All that water must've had a
*enormous* amount of momentum.  To guess at the momentum
that she was able to stop, you might estimate that the
volume of the flood was at least 100 by 100 by 100 to
1,000 feet ( 10^6 to 10^7 feet^3 of water ! ), and it
was moving at 50 or 60 miles per hour.  ( Think about
how tall the dam was and how much water was behind it. )


[ 2 ]  In 'Frozen,' Anna and Kristoff fell 100 to 200
feet off a cliff when they were trying to get away from
Marshmallow, Elsa's giant snowman.  What was their
velocity when they hit the snow below?


[ 3 ]  In 'Frozen,' Sven leaps a chasm ( with Anna on
his back ) to escape the pursuing wolves.  This may
seem implausible, but in fact reindeer can run 50 miles
per hour (mph).  Since Sven is carrying Anna, let's
call it 45 mph, or 66 feet per second (fps).  If we try
different angles at which Sven takes off to calculate
their trajectory, we'll see that Sven can indeed clear
a wide gorge.  If the initial velocity and angle are v0
and theta, and assuming they're in free fall after they
take off, their trajectory as a function of time is :

   x( t ) = v0 * cosine( theta ) * t

   y( t ) = ( v0 * sine( theta ) * t )
            - ( 1/2 * g * t * t )

where t is time, and g is gravitational acceleration
( 32.2 feet per second squared ).  x is along the
horizontal direction and y is in the vertical, with
+y = up.  Define the point of takeoff as x = y = 0.
An initial angle of about 15 degrees or more appears
to give the best results, in terms of clearing the
greatest distance - 60 or 80 feet or more.  But as
the angle increases, the initial velocity would
probably be less, since they're running more uphill
before taking off.

On the other hand, Kristoff would start decelerating
as soon as he cuts the sleigh loose from Sven, due
to friction, but it looks like he helped himself by
kicking off from his sleigh.


[4] In the beginning of 'Frozen,' the ice harvesters
load a large sleigh with blocks of ice.  It's about
4' wide by 6' high by 8' long ( or longer ), so
there's about 192 cubic feet ( ft^3 ) of ice.  Water
weighs 62.4 pounds/ft^3, and ice is 0.919 times as
dense as water, so the ice weighs about :

   192 ft^3 * 62.4 lb/ft^3 * 0.919
      = 11,010 lb
      = 5.51 tons

... which seems like a lot for two horses to pull.