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Would the Star Squish the Elf?

Lauren Anderson

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Introduction

Topic: https://youtu.be/3lTdLqjzWEY?t=205

Question: How much force is required from the blimps and the elves to stop the star, and could this force be overcome to move the star back up? Is this realistic for the bungee cords and the blimps?

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Methodology

  • Needed to use momentum equation to figure out how much force the star would have at ground level

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Δp = Δt * F

F = Δp / Δt

F = Δmv / Δt

F = mvf - mvi / Δt

F = mvf / Δt

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Methodology

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  1. Figure out height of tree (distance that the star fell)
    1. Height of elves compared to height of boy
      1. Average height of an 8 year old is around 1.3081 meters, and since two elves are almost the exact height of Billy, the elves would be about .65405 meters tall.

* this image is a screenshot to get more precise measurements, but that means it’s a little blurry- sorry!

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Methodology

  1. Height of tree cont
    1. Each floor of a typical building is approximately 4.2672 m and the average height of an adult male is 1.7526 m
    2. Each story = 2.44 * avg male height
    3. Building arrow is pointing to is 3 stories tall
    4. 2.44 * height of elf = 1 story
    5. 1 story * 3 = height of building
    6. Building = 4.7774 m tall

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Methodology

  1. Height of Tree Cont
    1. Set up a proportion using heights of the building and the tree measured from the screen vs. converted heights
    2. Tree = 22.7388 m tall
  2. Final velocity of star
    • Vf 2 = Vi 2 + 2 (a)(ΔY)
    • Vf 2 = 2 (9.8) (22.73888)
    • Vf = 21.11 m/s
  3. Time for star to fall
    • Vf = Vi + (a)(t)
    • 21.11 = 0 + (9.8) (t)
    • t = 2.15 seconds

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Methodology

4. Find area of star (to find mass) � A. First used the building height on screen, star height on screen, and the converted building height to set up a proportion � B. Star height = 1.812 m� C. Area of a pentagram can be divided into 10 triangles � D. Can find angles of each side using geometry � E. Long side = ½ of star height or .9061 m� F. Use Law of Sines to find another side

.9061 / sin (128) = x / sin (18)

x = .355 m

36

18

128

H = .9061

x

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Methodology

4. Area Cont. � A. Area = ½ (h)(x)(sin 36)� B. Area = .09455 m2� C. Area * 10 triangles = .9455 m2�5. Area to Mass � A. The average size of a tree topper is 4 in by 4 in (10.16 cm by 10.16 cm)� B. Using same steps as above, area = .00065 m2. To � C. How many tree toppers fit inside of giant one? � .9455 m2/ .00065 m2 = 1,455 � D. Mass of average tree topper = 1.32 lbs (.599 kg) � E. .599 * 1,455 = 871.52 kg

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36

18

128

H = .9061

x

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Methodology

6. Finding the force � A. F = mvf / Δt� B. F = (871.52) (21.11) / (2.15) � C. F = 8,540.9357 N

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Results

  • Each elf and bungee cord would have to sustain a force of 2,846.98 N (290.51 kgs) each in order to successfully bring the star back up
  • bungee cords must provide a minimum breaking strength of 1,500 lbs or 680.38 kilograms (6,667.724 N)
  • A typical blimp can hold up to 50 tons, or 45359.237 kg (444,520.52 N)
    • 2 blimps present, double this weight can be sustained

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Conclusion

  • the star is falling with a force of 8,540.9357 N
  • A single blimp can sustain 444,520.52 N
  • Each bungee cord can sustain at least 6,667.724 N
  • the bungee cords and the flying blimps are more than capable of sustaining the weight and stopping the star from hitting the elf on the ground.
  • The contraption should also be able to overcome to force of the falling star and lift it back up

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References

“Guide to Christmas Tree Toppers.” Balsam Hill, https://www.balsamhill.com/how-to-choose-the-right-christmas-tree-topper.

“How Tall Is a Storey in Feet?: Storeys to Feet.” Skydeck Chicago, 7 May 2021, https://theskydeck.com/how-tall-is-a-storey-in-feet/#:~:text=The%20height%20of%20each%20storey,average%20of%20about%2014%20feet.

Kim, Soo. “New York City's Rockefeller Center Christmas Tree and Star-Height and Weight Revealed.” Newsweek, Newsweek, 1 Dec. 2021, https://www.newsweek.com/rockefeller-center-christmas-tree-height-star-weight-1654826#:~:text=Featuring%2070%20spikes%20covered%20in,architect%20Daniel%20Libeskind%20in%202018.

Magazine, Smithsonian. “An Airship the Size of a Football Field Could Revolutionize Travel.” Smithsonian.com, Smithsonian Institution, 7 Mar. 2014, https://www.smithsonianmag.com/innovation/how-an-airship-the-size-of-a-football-field-could-revolutionize-air-travel-180950007/#:~:text=Unlike%20planes%2C%20it%20can%20take,globe%E2%80%94twice%2C%20creators%20say.

“Normal Growth.” Normal Growth Rates for Kids | Children's Wisconsin, https://childrenswi.org/medical-care/adolescent-health-and-medicine/issues-and-concerns/adolescent-growth-and-development/normal-growth.

Stone, Greg. “Bungee Cord Specifications.” SportsRec, 15 Oct. 2019, https://www.sportsrec.com/bungee-cord-specifications-7561664.html.

“What Is the Average Height for Men?” Medical News Today, MediLexicon International, https://www.medicalnewstoday.com/articles/318155#average-height-for-males-by-region.