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Can a Helicopter Lift a Ship Full of Gold?

Nathan and Sully looking for lost treasure— $5 billion in gold
Treasure found in two pirate ships
Ships are lifted individually by two helicopters

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Sikorsky S-64 Skycrane (CH-54 Tarhe)

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Momentum Theory

v₂ = (v₀ + v_e)/2

v₀ = 0 —> v_e = 2v₂

T = ṁ(v_e - v₀)

ṁ = ⍴Av₂ —> T = ⍴Av₂(2v₂) = 2⍴A(v₂)²

P = ½ ṁ(v_e)² = ½ ⍴Av₂(2v₂)² = 2⍴A(v₂)³

v₂ = (P/2⍴A)^⅓

T = 2⍴A((P/2⍴A)^⅓)² = (2⍴AP²)^⅓

Proposed in 1865 by William Rankine
Follows Bernoulli’s continuity principle

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Thrust Equation

T = (2⍴AP²)^⅓

T: Thrust

⍴: Air density = 1.225 kg/m³

A: Disk area = 378.25 m²

P: Engine power = 4,500 hp ≈ 3,355 kW

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Thrust Calculation

T = (2 * 1.225 * 378.25 * 3355000²)^⅓

T = 218495.93

Thrust: 218,495.93 N (49,120 lb)

Subtract helicopter weight:

218495.93 - 87629.97 = 130865.96

Max load: 130,865.96 N (29,420 lb)

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Weight of Gold

Current gold price: $59,284.60 per kilogram

Gold value in movie: $5 billion

Mass: 5000000000 / 59284.60 = 84,338.93 kg

Mass per ship: 84338.93 / 2 = 42,169.47 kg

Weight: 42169.47 * 9.8 = 413,260.78 N (92,905 lb)

Three times max load

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What if they used a more powerful helicopter?

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Mil Mi-26

Rotor area: 804.25 m²

Power: two 8,500 kW engines

Max load: 196,000 N (44,000 lb)

Less than half the gold weight

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Conclusion

The helicopter used in the movie is not capable of lifting the gold
There is no helicopter on the planet that can lift that amount of gold

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Works Cited

“Erickson S-64F/CH-54B Aircrane.” Vertipedia, https://vertipedia.vtol.org/aircraft/getAircraft/aircraftID/206.

Guide, Aeronautics. “Helicopter Aerodynamics of Flight.” Aircraft Systems, Blogger, 11 Apr. 2022, https://www.aircraftsystemstech.com/2017/06/helicopter-aerodynamics-of-flight.html#:~:text=Forward%20Flight&text=In%20straight%2Dand%2Dlevel%2C,than%20weight%2C%20the%20helicopter%20descends.

“Mil Mi-26.” Wikipedia, Wikimedia Foundation, 11 May 2022, https://en.wikipedia.org/wiki/Mil_Mi-26.

“Momentum Theory.” Wikipedia, Wikimedia Foundation, 9 June 2017, https://en.wikipedia.org/wiki/Momentum_Theory.

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Works Cited (Cont.)

“Monex Live Gold Spot Prices.” Monex, 30 Mar. 2022, https://www.monex.com/gold-prices/.

“Pratt & Whitney T73.” Wikipedia, Wikimedia Foundation, 27 Apr. 2022, https://en.wikipedia.org/wiki/Pratt_%26_Whitney_T73.

“Sikorsky Archives: S-64.” Sikorsky Archives, https://www.sikorskyarchives.com/S-64_Product_History%20modX.php.

“Sikorsky S-64 Skycrane.” Wikipedia, Wikimedia Foundation, 24 Feb. 2022, https://en.wikipedia.org/wiki/Sikorsky_S-64_Skycrane.