Initial Analysis 

A quarter would sit on top of a spring that is initially in equilibrium. The spring is compressed by a fluid being heated inside a chamber. Then, allow pressure to build in the chamber again. Lastly, release the pressure instantaneously shooting the quarter 2 meters into the air. 

Some potential fluids we can use for the chamber include air, water, and alcohol. We wanted to find the perfect fluid to use, one that has a low boiling point and low specific heat capacity.

Boiling Point & Specific Heat Capacity 

Air = x°C, 1.020 J/g°C

Water = 100 °C, 4.184 J/g°C

Alcohol = 78.37 °C, 2.460 J/g°C

Looking at the above numbers we decided it would be best to use air since it is already in the gas phase, and it has the lowest specific heat capacity and it is readily available.

Find: Q

Knowns - Gas:

K (spring constant) = 1N/m

Energy From Candle = 233 J

Spring potential energy: U = 1/2kx^2

233 = ½(1)(x^2)

So the displacement of our spring has to be 15.26cm. This is a very large displacement. So this idea might not be as feasible, because of the energy required to return the spring to its equilibrium position.​​

A Stirling engine uses the expansion and compression of a gas to create movement or work. There is a closed cylinder with a constant volume of gas. A heat source, such as a candle, heats the cylinder which expands the gas. This expansion moves a piston until the piston reaches the heat sink. The purpose of the heat sink is to absorb the heat, so that the gas can cool and compress, thus creating a cycle of expansion and compression. The most efficient Stirling Engines work based on a large temperature gradient.

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Mass of quarter = 5.67 g

Weight of quarter = (.00567 kg) * (9.81 m/s^2) = .0556 N

Work to move quarter up 2m = (.00567 kg) * (9.81 m/s^2) * (2 m) = .1112 J

To lift quarter up, stirling engine must produce a lifting force greater than .0556 N

The force the engine can produce will depend on the heat source, gas, the cylinder size, and the weight of the piston and flywheel. The heat source is the candle, and for gas, we plan to use air, unless we can get a hold of a gas with a lower specific heat than that of air, since then it would expand quicker. We do not want a large cylinder since a larger volume of gas takes longer to heat up. Force of the engine will be F = ma, where m is the mass of the piston, and a is the acceleration of the piston, or in other words, the expanding gas. We do not have the values for the cylinder size, the volume of gas, or mass of piston, but we can derive formulas to calculate the force the engine produces in terms of q, the candle heat.

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F_engine = Fexpansion - friction

Friction = (μ static) * (piston mass) * (g)

F_expansion = pressure * Area

Pressure = RT/V

q = m * c * (delta T) c = 1.020 J/g°C

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Before the piston can move, temperature must raise high enough so that force of the pressure breaks the frictional force of the piston.

By composing a cannon like a barrel containing a specific fluid, we can rest a quarter inside the cannon which ultimately acts as a cannonball. Where the fluid inside the barrel would be heated by the candle and build up the pressure. Once enough pressure is built, then the coin would launch into the air by flicking the cannon the pressure that built inside would shoot the coin two meters in the air vertically. This method is very similar to the spring method, but the quarter is placed directly above the pressure chamber instead of on top of a spring. 

The below calculations show how much force is required to launch the quarter. 

ΣF = Fqcosϴ - Fg = 0

Fq must be much greater than Fg in order launch the coin

Mass of quarter = 5.670 grams = 0.00567kg

Fg = m*g =  (0.00567kg)(9.8 m /s2) = 0.055566 N

If placed at ϴ = 20°

Fg = Fqcosϴ →0.055566 N = Fqcos(20) → Fq =  0.055566 N/ cos(20) → 0.59132N​

Using → F = ma,  (where we need to find acceleration to know F) we can find the force.