**The
“Safest” Roof to Fall Off of**

May 2017

*Table of Contents*

** **

__Background__:

One of mankind’s most basic necessities comes in the form of shelter: a roof over one’s head. Although roofs in general not only provide a person safety from the weather, environment, animals, and various other threats, they also allow the builder or owner self-expression in the form of architecture and style.

While originally made with nature, roofs have evolved greatly over time and developed many styles most suited for optimal living conditions or for just stylistic purposes. For example, New England Cape Cod houses were traditionally very boxy with a steep, shingled pitch, or roof slope, and a central chimney to not only retain heat in the moderately harsh New England whether, but also because it was a cheap house to make. Nowadays, New England Cape Cod houses can be found with expanded wings, two stories, stone roofing, and many other features.

Ultimately, roofs are built with the primary purpose of keeping things from building up on the house, regardless of roofing material and pitch. Although certain material and pitches prove to be beneficial in disposing things, like rain or tree debris, there is always the risk of someone getting injured when on or around a roof.

Although it is very rare for a projectile to slide off a roof and hit someone in the process, it is not uncommon for a person to fall off one and get hurt, even die in some cases. For example, as cited by the Bureau of Labor Statistics of the U.S. Department of Labor, 800 workers in the construction industry died from falling-related incidents out of the 4,836 in 2015, with nearly half of those falls confirmed from at or below standard two story building heights.

__Statement of the Problem__:

Every year, hundreds of people fall -
and even die - from roofs related incidents around the world. In order to
prevent this statistic from growing, certain calculations should be validated
in order to persuade safety over style in some architecture. So, in this
investigation, the question follows: __based on house height, roof pitch, and
distance from the edge, which fall produces the least energy upon impact?__
By calculating these factors, it could be determined which housing styles
should be encouraged in order to reduce fall-related injuries.

__ __

__Hypothesis__:

Depending on whether a person would
rather prefer to go a distance or hit the ground harder instead, __I
hypothesize that the smaller house will result in lowest impact energy at all
pitches and distances compared to the taller house__. This is because I
believe gravity has a more significant contribution than velocity in this
particular situation.

** **For this investigation,
several steps are taken in order to calculate the final impact energy generated
after falling off a roof.

Before the beginning of the actual investigation, several basic steps are taken to convert several united, like pitch and feet, into angles and meters that better suited for calculations of complex physics equations. In this case, the independent variables are the height of the house, distance from the edge of the roof, and roof pitch; two different heights were used, ten and twenty feet which represent one and two story houses, distances from the edge were determined to be one to eleven feet, and pitches ranged from 15° to 85°. Furthermore, a solid sphere that has a mass of 80.7 kg was used to represent an average falling human as it allows for a very uniform shape for travel at the average mass of an American, both which would produce some room for error by producing generally higher results.

First, the
velocity accumulated through roof travel calculated in order to later calculate
the generated impact energy. To do this, the height in meters on the roof is
calculated by multiplying the distance from the edge by the sine angle of the
pitch. Next, the rotational velocity of a sphere down the roof is calculated
with the equation mgh = 0.5m(u_{x}^{2}) + 0.5(I)ω; this
then reduces to the equation u_{x} =, with a gravity of -9.81 m/s^{2} and a
height in meters based on the result of the aforementioned calculation.

From this point,
there are now two different ongoing equations at both ten feet and twenty feet
heights as gravity now plays a primary role in generating additional velocity.
This is used in order to simulate potential results from common one and two
story houses in America. In order to calculate the velocity generated by
gravity, the rotational velocity generated by roof travel primarily affects the
result; the equation then follows: u_{y} = . From then, the final impact velocity in meters per
second is calculated as a vector of the rotational velocity and velocity
generated by gravity, with v = .

Now, with nearly all
calculations needed, the energy generated upon impact can be determined as a
vector from gravitational velocity and acceleration generated from the fall. In
order to get a better scope of the impact, the energy will be calculated at a
depth of 5 cm. Gravitational energy is calculated by multiplying the mass by
gravity and depth below the earth’s surface, which equals to -39.583 N.
Afterwards, the energy generated by the acceleration is calculated by squaring
the final velocity at twice the depth, then multiplying it by the weight making
the equation read F_{acceleration} = m(V^{2}/2s),simplifying
to, F_{momentum} = 807(V^{2}). Ultimately, this makes their
net, the impact energy in newtons, equal to F_{net} = 807(V^{2})
- 39.583.

*Pictured
above are visual representations and estimations of what the investigation
would look like and how it would be calculated.*

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* *

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** **__As validated by
the results, the smallest house, pitch, and distance from edge does produce the
smallest impact energy, with 50,042.01758 N__. __Similarly,
if one analyzes the data trends, curves can group similar results, producing
information house size is the most contributing factor in calculate impact
energy__. For example, many of the most extreme variables of the smaller, one
story house produce results found at many of the smallest variables of the
larger, two story house.

From this information, several conclusions can be made in order to promote higher safety standards. For instance, water park designers can utilize this information in order to construct safer rides while still retaining the same levels of fun. Moreover, city planners and architects can utilize this information greatly in order to determine safe protocols when building new houses; for example, at a certain pitch, tree branches above the house would have to be cut in order to prevent future damages outside of the direct property. Finally, certain safety protocols can be established for working on certain architectures to ensure the liveliness of workers.

While this investigation can provide great insight at characteristics of safe roofs, there are several limitations that may skew actual results. For example, one large aspect is the neglection of friction and the assumption that a person wall travel like a sphere. In reality, both roofing and air friction would reduce generated velocities overall, producing lesser results. Additionally, when humans fall, they tend to flail and not travel and roll in a uniform fashion, like a sphere, so different results would be produced as well. Finally, the largest limitation of this investigation is that this is all hypothetical; in reality, it is very dangerous for one to actually jump off various roof-like settings to achieve actual results without injury.

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__APP #1__: “initial
calculations”

__APP #2__: “calculating
velocities from a 10 ft. house”

__ __

__APP #3__: “calculating
velocities from a 20 ft. house”

__APP #4__:“calculations from a
10 ft. house”

__ __

__APP #5__:“calculations from a
20 ft. house”

Morgan-Harlow, Cate. "Roofing Styles: Trends
and History." *BuildDirect Blog Life at Home*.

BuildDirect, 17 May 2012. Web. 5 Jan. 2017.

* *

*NATIONAL CENSUS OF FATAL OCCUPATIONAL INJURIES IN
2015*.
Washington, D.C.: U.S.

Department of Labor: Bureau of Labor Statistics, 16 Dec. 2016. PDF.

*Misc: Related
Websites*

*https://www.builddirect.com/blog/roofing-styles-trends-and-history/*

** **Value: Good site that goes over the architectural features of
popular roofs in the USA.

*https://www.bls.gov/news.release/pdf/cfoi.pdf*

** **Value: Reputable website source filled with statistics,
including roofing incidents.

* https://www.osha.gov/doc/engineering/EXengrptsr.html*

** **Value: Reputable website that notes other major construction-related
incidents as well.

*http://www.roofingcontractor.com/articles/91572-navigating-workplace-safety-issues-in-2016*

** **Value: Popular site amongst the construction community that
reviews 2016 and the safety issues faced and addressed.

*http://www.nclabor.com/news/2017/2017-01-30OFIR2016.pdf*

** **Value: Recent statistics in the industry after safety
reforms.

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