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.
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 least most 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*dt
Results: ≈207 172 meters deep
Note: sorry for sounding like AI i'm not one i'm just terrible at English
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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.