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HOW TO USE THIS TOOL
Snowman and -ball tool (Free version) (V0.03 Mar 2018)
Date of writing of this infotext: 29.03.2018
STRUCTURE
1) Disclaimer
2) Basic functions
--- 2.1) Calculations of volume and mass
--- 2.2) Calculations of the principal formation of the individual balls
--- 2.3) Calculations on the rolling of the balls/snowmen
--- 2.4) Calculations on the building of snowmen (as compared to -balls)
--- 2.5) Taking into account physical strength
--- 2.6) Calculations with the snow coefficients
--- 2.7) Tracking your snowman-building progress
--- 2.8) Calculations on the melting time
3) Complex functions
--- 3.1) Graphing variation of volume and mass
--- --- 3.1.1) Further variations among snowmen: Physics
--- --- 3.1.2) Asymptotic behaviour of snow on inclined planes
--- 3.2) Snowballs over the complex plane: Approximating highly random behaviour
--- 3.3) Practical advice for the user (saving calculation time)
--- 3.4) Energy-effective snowbuilding and its advantages NEW WITH VIDEO LINKS
4) Basis of the calculations
--- 4.1) Academic sources
--- 4.2) Own experiments
--- --- 4.2.1) Setup of the experiments
--- --- 4.2.2) Conducting the experiments
--- --- 4.2.3) Results and evaluation
--- --- 4.2.4) Disclosure form and licensing agreement (research results)
--- 4.3) Further reading (non-academic)
5) Summary of this infotext
6) Personal note
--- INFOTEXT START ---
1) DISCLAIMER
This software has been written in order for snowbuilders all around the world to have a one-stop all-included tool, with a special focus on snowman and -ball-building. To the best of the creator's knowledge, the formulae and their applications are correct, both mathematically and practically. However, one must acknowledge, that snowbuilding is a particularly dangerous past-time, and the creators will not come upfor any damages incurred through the use of this tool. Even the case of death and/or mutilation, no reaction shall be expected from the creators -- however, family members might be reimbursed with up to the full price of the software (provided that it has been bought regularly.) Local and international laws apply to varying extent.
2) BASIC FUNCTIONS (These are important)
2.1) CALCULATIONS OF VOLUME AND MASS
The only value asked off the user (you) is the diameter of the snowball. This equates to the height of the ball. In the case of a snowman-preset, the full height of the snowman is taken. It is very easy to calculate the volume of a sphere (ball) from the diameter: according to Formula 1:
V = 4/3 * pi * r^3 (Formula 1)
Where r is the radius. The radius is half the diameter.
More complicated formulae exist to extrapolate the volume of a snowman from its height, these will be discussed in more depth in Section 3.1). At this stage, it suffices to say that from each consecutive ball making up the snowman, a geometric series can be formed. By making an educated guess at the ratio, one can deduce the volume of the bottom ball and hence the volumes of each following ball.
Mass is a quantity that can be derived from volume and mass, as seen in Formula 2. Here, Volume is capital V and density is the lowercase rho (p in this formula):
m = V * p (Formula 2)
Based off the snow coefficients and real-world data, which will be discussed in greater depth in Section 2, the density of snow can be calculated.
Real-world link: Bigger balls are heavier.
2.2) CALCULATIONS OF THE PRINCIPAL FORMATION OF THE INDIVIDUAL BALLS
When creating a snowball, one begins by forming a small ball with the hands. This ball is then rolled in snow for it to gain thickness. The size of the starting ball depends on the make-up of the snow and the strength one can exert on it with two hands. Following formula has been found:
Diameter Initial Ball= 0.03*ln((strength/600)+4)*snowcoef (Formula 3)
Where ln() is the natural logarithm.
Real-world link: The stronger your hands are, the larger the snowballs you can press with them.
2.3) CALCULATIONS ON THE ROLLING OF THE BALLS/SNOWMEN
Now that you have the size of your initial snowball, you need to roll it in order for it to increase in size. The program finds out how thick the snowsheet is that you have to add to the snowball in order to reach the desired final diameter. Then it calculates the mass, as seen in Formula 4:
Mass of snowsheet = Mass final ball - Mass Initial ball (Formula 4)
The mass can be derived from the distance the ball is rolled. The formula (Formula 5) is quadratic as to represent the non-linear behaviour a snowball's mass experiences as it is rolled (the heavier a ball is, the faster it will gain weight).
Mass = (Distance^2)*snowcoef/4 (Formula 5)
One can solve this for the distance rolled:
Distance = 2*sqrt(mass)/sqrt(snowcoef) (Formula 6)
Real-world link: If you roll a snowball, its mass increases.
2.4) CALCULATIONS ON THE BUILDING OF SNOWMEN (AS COMPARED TO -BALLS)
There are three presets in this tool: Ball, Small Snowman and Big Snowman. Whereas the Ball contains only one sphere, whose diameter takes up the entire height of the structure, Small Snowman and Big Snowman have two or three balls respectively.
Small Snowman has two balls in a size ratio of 1:0.66. Taking into account that there is an overlap between the spheres of a snowman (to make sure that the contact area is large enough to provide sustained stability for the snowman), the formula is:
Volume = (4/5)*(4/3)*pi*((2/10 * h)^3 + (3/10 * h)^3) (Formula 7).
Where h is the total height of the construction. As one can see, the contact areas are expected to take up 40 percent (or 2/5) of the entire volume. Since they only need to be counted once (instead of twice), the volume is reduced by 20 percent (or 1/5), hence, the volume is multiplied by 4/5, which has the same effect.
Quite similarly, Big Snowman has three spheres with a size ratio of 1:0.75, meaning that the third sphere will have a size of 0.75^2 :1 (compared to the largest, bottom, sphere). Hence:
Volume (4/5)*(4/3)*pi*((0.12^3)*h + (0.16^3)*h + (0.43^3)*h) (Formula 8)
Again, h is the total height of the construction. One can see that the volumes of the constructions will always be less than the volume of a sphere of the same size -- which makes sense, since the sphere does not contain the empty space included in the snowman constructions.
Real-life link:
2.5) TAKING INTO ACCOUNT PHYSICAL STRENGTH
Your physical strength is very important in snowbuilding. With greater strength comes the ability to build larger structures in less time. In mathematical terms:
Time in seconds = 3*(d*20 - f/15) (Formula 9)
Where d is the distance in meters and f is the force exerted by the body parallel to the ground (normal to the acceleration due to gravity). There exist variations to this formula, which take into account the intrinsic aspects of sloped planes, which will be discussed in further depth in Section 3.2).
From a variety of sources, the creators have concluded that the energy in kcal (kilocalories) used up in the process of building snowmen lies at roughly 300kcal per hour. This energy is largely expediated as a result of moving around snow objects of great weight, some more energy is lost to keep the snowbuilder warm (snowbuilding is an outdoor sport). See formula 8 for the calculation method from which the energy used up in snowbuilding is derived.
Energy in kcal = t*0.08333*f (Formula 10)
Where t is the time in seconds and f is the efficiency (an integer from 1 to 4 representing the body's efficiency in energy use).
Real-world link: It takes time and energy to snowbuild.
2.6) CALCULATIONS WITH THE SNOW COEFFICIENTS
The make-up and usability of snow varies with temperature and graining. Higher temperatures make snow stickier, but at the same time the chance of melting is much increased. Coarse-grained snow is better for snowbuilding due to its increased stickiness. In order to weight them properly, the following formula has been derived:
Snow Coefficient = (grain - temp)/2 (Formula 11)
Where temp is temperature in degrees Celsius and grain is an arbitrary scale from 0 to 10 (10 being coarsest). The formula does not work very well above 0°C, since it does not accurately reflect how any snowbuilding becomes impossible when the snow has melted.
Real-world link: Bad snow can mean that your snowman will be impossible to build.
2.7) TRACKING YOUR SNOWMAN-BUILDING PROGRESS
You can choose to record building progress. If you enable it, the software will save the distance you had to roll your ball(s). Since we are no major internet company from California, we cannot afford servers to backup your data, so please keep your devices safe or back your crap up yourself.
However, we can create a histogram for you, depicting the frequencies of the different distances we predicted you had to roll your snowballs.
2.8) CALCULATIONS ON THE MELTING TIME
Everything built by us snowbuilders is prone to melting, as temperatures will eventually rise above 0°C. In the tool, the standard thawing temperature has been chosen to be 10°C. The assumption is that the temperature is constant and that energy is applied evenly onto all parts of the body. Sadly, this is not the case (night, duh, shadows, duh). So in order to take into account some of the environmental factors, six factors have been chosen: Shade, Slope, Distance to Houses, Distance to Forest, Windiness, Sunglasses.
The formula gives the melting duration for the entire construction to turn to water in days.
Melting time in days = (v -t*20)*((a+1)/3)/180 (Formula 12)
Where v is the volume of the construction in m^3, t is the temperature in degrees Celsius and a is the sum of all environmental factors (between 0 and 6).
However, even a slightly thawed/molten snowshape can lose its appeal and beauty since detailing is the first thing to be destroyed in the melting process. The creators do not guarantee that a partially molten snowshape even resembles its initial form.
The best temperature to build a snowman is -1°C.
Some instructions on building a snowman have even been patented.
One of the first forms of folk-art known to man!
A snowman should have a foundation of at least two inches of wet snow.
Snowmen are made of snow!
Whats the difference between snowmen and snowwomen? Snowballs!
Michelangelo made one!
The largest snowman built was approximately 37 meters high
In Switzerland, spring is celebrated by blowing up a snowman.
Japan holds the world record for building the most snowmen in 1 hour.
Karen Schmidt owns the largest collection of snowmen items in the world.
Scientist made a snowman that is just 0.01mm tall.
By building a snowman, you burn about 238 calories per hour!
Frosty the snowman!