A classical mechanics problem by Nazmus sakib

Classical Mechanics Level 2

Selah dropped a stone into a deep well. After 3 seconds, she heard it hit the water.

If the speed of sound is $\SI[per-mode=symbol]{300}{\meter\per\second}$ , determine the depth to the surface of the water. Give your answer in meters to 1 decimal place.

Take $g = \SI[per-mode=symbol]{9.81}{\meter\per\second\squared}.$

The answer is 40.3.

2 solutions

Brian Charlesworth
Mar 25, 2017

Letting the depth of the well be $d$ , the time for the sound to travel back to the top of the well would be $\dfrac{d}{300}$ seconds, so the stone actually takes only $3 - \dfrac{d}{300}$ seconds to reach the water. The equation of motion of the stone, with an initial velocity of $0$ m/s and with $g = 9.81$ m/s $^{2}$ , will then be

$d = \dfrac{1}{2}g\left(3 - \dfrac{d}{300}\right)^{2} \Longrightarrow 90000d = 4.905(900 - d)^{2} \Longrightarrow$

$90000d = 4.905(810000 - 1800d + d^{2}) \Longrightarrow 4.905d^{2} - 98829d + 3973050 = 0$ .

Using the quadratic formula we find that $d \approx \boxed{40.3}$ meters.

Can you solve it more clearly?

Nazmus sakib - 4 years, 2 months ago

I've added a few steps for clarity. Upon recalculation I found the answer was $40.3$ meters to 1 decimal digit rather than the $40.2$ meters I found when I first made the calculation, but I think that anyone entering $40.3$ as their answer will still be considered correct as it constitutes just a 0.25% discrepancy.

Brian Charlesworth - 4 years, 2 months ago

It seem that we are takeing a simple equation and makeing it very confusing.

Odell Drummer - 4 years, 1 month ago

The "simple" equation $d = \dfrac{1}{2}gt^{2}$ is still being used, but we have to make an adjustment for the time sound takes to travel from the bottom to the top of the well. Which part of my solution do you find confusing?

Brian Charlesworth - 4 years, 1 month ago

A Former Brilliant Member
Mar 24, 2017

That would be the answer if the speed of sound were infinite, but that is not the case, you need to consider that sound takes some time to go from the water surface to the top of the well. The 3 seconds include the time the stone takes to get to the water, plus the time sound takes to get to the top.

Luis Villalba - 4 years, 1 month ago

In my opinion, the answer must be 11,025m. Let me explain you my theory efficiently. -As we know the speed of sound is 300m/s(we consider it as a constant speed because air friction hasn't been taking in count in this situation, just for gravitational force hasn't been affected by air friction as well, because the acceleration due to gravity would have been diminished.) Following this direction, we can presume that time elapsed in this case represents the double of the time required for the rock to fall from his hand onto the surface of the water(sound has to travel from his hand to the water and afterwards rebound into the person's ear in order for him to hear it.) Since sound travels at a constant speed, the time required is proportional to the distance travelled in this case. Thus, since the total distance is constant in this case for sound, sound has taken precisely 3 seconds in total. We can also conclude that the time taken by the rock is 1.5 s(half of 3 s). -As the initial speed of the rock in y direction is 0m/s, the acceleration(given in this situation) 9,81m/s2, we can calculate the total distance taken by the rock to fall from the person's hand onto the water surface. Δy= viyΔτ+1/2αΔτ2. You can find the identity of Δy(distance between the ground and the surface of water, which is equal to the distance taken by the rock to travel onto the surface of water).

Jia Bo Cheng - 4 years, 1 month ago

Answe is 13.26

Sarada Peri - 4 years, 1 month ago

A classical mechanics problem by Nazmus sakib

The answer is 40.3.

2 solutions

0 pending reports