# Abyss

Stone was pushed into the abyss: 2 seconds after we heard hitting the bottom. How deep is the abyss (neglecting air resistance)?

(gravitational acceleration g = 9.81 m/s2 and the speed of sound in air v = 343 m/s)

Result

h =  18.6 m

#### Solution:

$v = 343 \ m/s \ \\ t = t_1 + t_2 = 2 \ s \ \\ h = \dfrac12 g t_1^2 = v t_2 \ \\ h = \dfrac12 g t_1^2 = v (2 - t_1) \ \\ g t_1^2 = 2 \cdot 2 \cdot 343 - 2\cdot 343 t_1 \ \\ 9.81 t_1^2 + 686 t_1 -1372 = 0 \ \\ \ \\ t_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ -686 \pm \sqrt{ 524433.28 } }{ 19.62 } \ \\ t_{1,2} = -34.96432212 \pm 36.910176236537 \ \\ t_{1} = 1.9458541162516 \ \\ t_{2} = -71.874498356822 \ \\ h = \dfrac12 g t_1^2 = 18.6 \ \text{ m }$

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