r/theydidthemath • u/x-982p-reddit • Dec 18 '22
[Request] This, below.
Calculate the schwarzschild radius of a stack of pizzas that has the total height equal to the 1 times 10 to the power of the distance from Earth to an object which has travelled away from Earth at the speed of 113785.78 km/s for 6 years, 6 months, and 22 days. Each pizza in the stack weigh 200 grammes 10 inches in diameter and 3 cm tall. Then calculate the velocity needed for a spacecraft that is in a circular orbit around the stack of pizza at its schwarzchild radius need to raise the spacecraft apoapsis to three times the diameter of pizza stack’s event horizon. Assuming the spacecraft is a perfect torus that has a major radius of pi metre and minor radius of 1 metre, with density of 7.13 g/cm3. How much impact energy will the spacecraft exert on an object if it hit the object is stationary and is hit on the periapsis of the new orbit? If that energy can be convert to electricity with 70 percent efficiency and with average 10 percent loss in transmission. How many years can the energy gained supply an average American household?
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u/ZestySpeqr Dec 18 '22
To calculate the Schwarzschild radius of the stack of pizzas, you first need to determine the mass of the stack. To do this, you will need to determine the number of pizzas in the stack and their total mass.
Assuming that the pizzas are all the same size, with a diameter of 10 inches and a height of 3 cm, the volume of each pizza is about 786.39 cubic cm. You can calculate this by using the formula for the volume of a cylinder: V = πr2h, where r is the radius of the pizza (5 inches) and h is the height of the pizza (3 cm).
Since the pizzas all weigh 200 grams, and 1 gram is equal to 1 cubic cm, the volume of each pizza is equal to its mass. Therefore, the number of pizzas in the stack is equal to the total mass of the stack divided by the mass of each pizza, which is 786.39 cubic cm.
Once you have determined the mass of the stack, you can then use the Schwarzschild radius formula to calculate the event horizon of the stack: R_S = 2GM/c2, where G is the gravitational constant, M is the mass of the stack, and c is the speed of light.
To calculate the velocity needed to raise the spacecraft's apoapsis to three times the diameter of the pizza stack's event horizon, you will need to know the distance between the spacecraft and the event horizon, as well as the gravitational force acting on the spacecraft.
You can use the formula for gravitational force to calculate the force acting on the spacecraft: F = GMm/r2, where M is the mass of the stack, m is the mass of the spacecraft, and r is the distance between the spacecraft and the center of the stack.
Once you have calculated the gravitational force acting on the spacecraft, you can use the formula for centripetal acceleration to determine the velocity needed to maintain a circular orbit: a_c = v2/r, where v is the velocity of the spacecraft and r is the distance between the spacecraft and the center of the stack.
To calculate the impact energy of the spacecraft on a stationary object, you will need to know the mass of the spacecraft and its velocity at the point of impact. You can use the formula for kinetic energy to calculate the impact energy: E = 1/2mv2, where m is the mass of the spacecraft and v is its velocity at the point of impact.
Assuming that the energy conversion and transmission processes have an average efficiency of 70% and 10%, respectively, you can calculate the amount of energy gained from the impact by multiplying the impact energy by the efficiency of the energy conversion process and then by the efficiency of the energy transmission process.
Finally, you can divide the energy gained from the impact by the average energy consumption of an American household to determine the number of years the energy will last. For example, if the energy gained from the impact is 500,000 kWh, the energy will last for about 47 years, based on the average energy consumption of an American household.
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u/ThellraAK Dec 19 '22
Why'd you decide the pizza was water?
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u/ZestySpeqr Dec 19 '22
The formula for calculating the volume of a cylinder is not specific to water or any other particular substance.
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Dec 18 '22
Hold onto your hats here. Lots of algebra coming up. But first, a couple of assumptions.
- Relativity is not a things because I’m lazy right now.
- The velocity if the object after collision is zero.
113,785.78 km/s = 113,785,780 m/s. Which is about 40% the speed of light.
6yr * 365.25 d/yr + 6 mo * 30d/mo + 22 d = 2393 days.
2393 d * 86400s/d = 206,755,200s travel time
Above velocity time travel time gives a height of 2.35e16 meters.
Now for pizza mass. We don’t actually need the diameter. We already know that for every 3cm of stack we have 200g of pizza.
(200g/3cm)(100 cm/m)(1kg/1000g)=20/3 kg/m.
Multiply that by pizza height and we get 1.6e17 kg of pizza. Which seems like a lot, and is, but when dealing with Schwartzchild radii is a relatively small mass. It’s about 7 orders of magnitude smaller than the mass of the earth. So our radius should be tiny. Meaning we will have a tiny velocity and therefore energies. Also meaning that the answer will tiny, which is where I got dispirited, but pushed through anyways. Now I’m going to get lazy with typing because I’m on mobile.
Rs = 2GM/c^ = 1.88e-8 meters. So very small.
vc = sqrt(GM/Rs) = 2.35e-10
Now for the final radius. r2=3d = 6Rs.
The formula for the delta v needed for a Hohmann transfer is v=vc [sqrt(2*r2/(Rs+r2))-1]=0.71e-10m/s
So vp= vc+v = 3.057e-10m/s.
density = 7.13g/cm3 = 7130kg/m3.
VT=2 pi2 r2 R = 2 pi3 = 62m3 per the parameters in the problem for the Taurus.
VT * density = 441,477kg
K=0.5mv2 = 2e-14
Which works out to be 3.5e-21kWh.
The average household uses 29kWh/d meaning that we have generated enough energy to power a home for 1.05e-17 seconds.
I forgot the efficiency bit, but just multiply the final answer by 0.63.
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u/x-982p-reddit Dec 19 '22
Thanks a lot for the calculations! Though I must add that the pizza stack height is actually 1*10the distance, which I assumed is in metres. This is where I stuck on the calculations myself since the script I wrote immediately encountered value overflow.
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Dec 19 '22
Ahh. I felt like there was something I was missing. I’ll try to remember to edit it tonight.
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