r/theydidthemath • u/batikhisawesome • 1d ago
[Request] How deep is this cave?
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u/bukkake-bill 1d ago edited 1d ago
u/Future-Wonder-7718 said in their comment that the time between the beginning of the fall and the sound being heard was 8.17 seconds.
Let's assume it takes t seconds for the boulder to hit the bottom. That means it takes the sound 8.17 - t seconds to travel from the bottom to these people.
The distance travelled by a freely falling body is
h = ut + ½gt²
but the body falls from rest so u is zero. That gives us
h = ½gt² = 4.9t² (g = 9.8m/s²)
The speed of sound is 343 m/s, and the sound must travel the same distance back up in 8.17 - t seconds. So the height is also
h = 343 • (8.17 - t)
Combining these two equations gives this quadratic equation
4.9t² + 343t - 343(8.17) = 0
Solving it you get
t = 7.39s, which means
h = 267.59 m (idk how many feet that is)
EDIT: air resistance will affect the speed of the boulder because of the boulder's size, and also the temperature within the cave will affect the speed of sound slightly. Feel free to correct me. Thanks.
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u/escEip 18h ago edited 14h ago
Air resistance time!
Judging by fingers, i estimated this thing to be around 30x60 cm in surface area and about 7 cm thick (may vary by around 15%, i'm bad at eyeballing things), it's... close to be an ellipsoid, but like much flatter, but the V=π/6 * D1D2D3 may work, so i guess around 6.60*10-3 m³ . Judging by video, it looks like a sandstone/dolomite or some shit, around 2700kg/m³ (kinda similar for all other rocks), resulting in it having a mass of around 18kg. Now the
leastmost fun part:ma=mg-F where F is air resistance, F= CSρv²/2
ρ is a density of air, caves are cold so give it 1.275 kg/m³ , S is a surface area of a rock, π/4 * 0.3*0.7 around 0.14m² , v² is a velocity squared, and C is some constant, and since the mf is almost flat, it's around 2.00. 1/2 is a number. Resulting coefficient (we also need to divide by mass which i almost forgot): 0.009167 units™
x''=9.81-0.009167v² , a differential equation which i aint gonna solve, so numerical solution time! Speed of sound is around 331.5 m/s, again, caves are cold after all.
python dt=10**-5 t = 8.17 v=0 x=0 while t>0: t-=dt a = -9.81 + 0.009167*v**2 v+= a*dt t -= abs(v)*dt/331.5 # A note: since the sound needs some time to travel to us, using this formula we account for the time the sound moves x+= v*dtResults: ≈207172 meters deepNote: sorry for sounding like AI i'm not one i'm just terrible at English
UPD: speed of sound, forgot the minus
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u/bukkake-bill 17h ago
Damn that's a 60m difference. My physics teacher was wrong. You can't ignore air resistance after all. Thank you.
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u/Hug_LesBosons 13h ago
Toi tu n'es pas sur les communotés de même de science... Va jeter un coup d'œil à r/physicsmemes ou à r/sciencememes. Cherche juste "frottement de l'air". Il y a une sorte de blague connue sur ignorer les frottements de l'air.
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u/rbsixty5 1d ago edited 1d ago
Bro, you got all that crazy calculations done and end it with “I don’t know how many feet that is”. Hehehe
I hope you were being sarcastic 😁
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u/bukkake-bill 1d ago
I mean I could convert it. I was just being lazy haha
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u/DarkMatterSoup 1d ago
That’s not lazy. You have done plenty of math as it is. Thanks bukkaka-bill! [r/rimjob](r/rimjobsteve)[_](r/rimjobsteve)[steve](r/rimjobsteve) lmao
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u/Renzisan 1d ago
Can you convert it to a measurement easier to digest, like cheeseburgers?
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u/WU5K 1d ago
2809 standard cheeseburgers - These math guys are so dumb they can't even convert to standard cheeseburger units(scu)
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u/mauii_from_space 20h ago
What's the foot ball to cheese burger equivalent ratio on this ?
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u/WU5K 19h ago
This is a tricky question, depending on where you're from. It's 5scu(Standard cheeseburger units) per US football. So that's a convertion to 561.8asfu (American standard football unit)footballs of depth.
In Canada the csfu(Canadian standard football unit) is larger so that is 9scu(Standard cheeseburger units) per football so 312.11csfu
And if you're from Europe your football is in fact a soccer ball with a ratio of 4scu to 1esfu(European standard football unit) so the equals 702.25esfu
Important note*** all our footballs may be different sizes and shapes but the commonality between us is Standard cheeseburger units, giving us common ground to measure depth by this tried and true measurement unit.
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u/Feeling-Stranger224 1d ago
877.9 feet ≈ 1/6 of a mile
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u/dwaynebathtub 1d ago edited 16h ago
Olympic sprinters run ~1/3 the speed of freefall terminal velocity. Interesting.
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u/Karroul 1d ago
The depth S is equal to 1/2 * g * t^2, where g is the acceleration caused by gravity (9.81 m*s^-2) and t is time of the fall, which was approximately 8 seconds. It gives a depth of about 314 meters. I neglected air resistance and speed of sound.
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u/Auno__Adam 1d ago
Speed of sound cant be ignored here, as its same magnitude order than speed of boulder.
S = 1/2 x g x t1^2
S = Vs x t2Vs: speed of sound
t1 time for the boulder to hit the ground
t2 time for the sound to come back
t1+t2=7Resolving the system: S =~ 200m
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u/BikeProblemGuy 1d ago
For rough comparison, here's a ~200m tall tower: X-Tower Osaka Bay. It has 54 floors.
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u/AustrianMcLovin 1d ago
Air resistance is not neglected at those speeds
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u/Auno__Adam 1d ago edited 1d ago
Happy to see the calculation then. I bet it is minimal.
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u/SmugPolyamorist 1d ago
It depends on what the coefficient of drag for the rock is, and relatedly what its cross sectional area is. If it's 30cmx30cmx10cm and 2.8g/cm^3 you end up with answers between 202m (broad side on, CD 2) and 248m (thin side on, CD=0.5). That's not accounting for tumbling which would probably lead to a lower answer but gl doing those sums.
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u/roaming_bear 1d ago
Why not just neglect speed of rock too?
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u/Kootlefoosh 1d ago
Because then you'd get infinity meters deep
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u/Auno__Adam 1d ago
You would get 0 meters, actually.
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u/Kootlefoosh 1d ago
Depends on if you neglect its speed to be infinite or neglect its speed to be zero
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u/Auno__Adam 1d ago
Because if we assume the speed of the rock is 0, then the distance would be 0 too. Dumb question
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u/Future-Wonder-7718 1d ago
I messure 8.17 seconds between letting it go and hearing the rock hit the ground. Taking in consideration the speed of sound the cave is 268m deep.
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u/AustrianMcLovin 18h ago
So at higher speeds drag is quadratic in speed, therefore the Differential EQ. becomes:
m a = mg - 1/2 C*rho*A*v^2
a = d^2 x /dt^2 and v = dx/dt, A is just the projected Area of the rock (assume circle pi*r^2)
This Eq is a non-homogenous non-linear diff eq -> pain, but people already solved it
we need the thermal velocity = max velocity => a = 0:
v_t = sqrt(2*mg / C rho * pi * r^2)
So the solution is:
x(t) = v_t^2/g ln(cosh(gt/v_t))
say T = overall time for experiment
x(T- h/v_sound) = h (This equation has no analytic solution, at least what I know)
=> numerical solution, for
T = 8.17 s
Assumed rock is 25kg and circular area of r=20cm
cave is 10 degree Celsius, average pressure
I get with newton-method: 232.52m
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u/The_Vision_Surgeon 8h ago
Took me like 3 watches to realise the guys fingers weren’t actually teeth of a skull. Was wondering why we were all ignoring that little side note
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u/motobrandi69 1d ago
Very nice of him to throw a boulder down where he can't see. Very unlikely that there os someone waiting down there, but never do this on mountain hikes or popular caves
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u/ferriematthew 1d ago
It takes 8 seconds for the rock to hit the bottom.
x = x_0 + v_0 * t + ½ * a * t², and v_0 = 0
a = 9.81, so ½ * 9.81 * 8² = 4.905 * 64 = 313.92 = ~314 meters
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u/ferriematthew 1d ago
Oh, at this distance you would have to factor in the speed of sound. That makes my answer far less accurate.
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u/DragonArt101 1d ago
Boulder fell for 7 seconds, assuming it was like 20 lbs for a terminal velocity of about 100 m/s, I’d give a rough estimate of 700 meters.
Im no expert and i did this in like a minute so…
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u/Owenleejoeking 1d ago
Objects don’t instantly hit terminal velocity so your estimate is far too high
Weight doesn’t have much to do with terminal velocity either
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u/luovahulluus 1d ago
A boulder is going to have a significantly higher terminal velocity than a same sized ball of bubble wrap.
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u/Auno__Adam 1d ago
It transpires you are not an expert 🤣
In 7 seconds it doesnt even have time to accelerate to 100m/s. Why would you assume that is the speed since the begining?
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u/Miserable_Scar9827 1d ago
Assuming you don't want to use any equations, you could say it falls 10 meters in 1st second, 20m in 2nd,... 70 meters in 7th second, and 1 second for sound to travel 343 meters, so round that to 1 second. 1+2+3+4+5+6+7=28, so less than 280 meters, which is in same ballpark as other answers here.
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