Question

6 multiple choice 1 point a stone is dropped into a deep well and is heard to hit the water 3.41 s after being dropped. determine the depth of the well. (assuming the sound started right when splash happens and it took 3.41 s to hear sound) 114 m 11.63 m 57 m 0 m clear my selection

Explanation:

Step1: Define variables

Let the time taken for the stone to fall be $t_1$ and the time taken for the sound to travel back be $t_2$. The total time $t = t_1 + t_2=3.41\ s$. The depth of the well $d$ can be expressed in two - ways. For the falling stone, $d=\frac{1}{2}gt_1^{2}$ (using the equation of motion $x = v_0t+\frac{1}{2}at^{2}$, with $v_0 = 0$ and $a = g=9.8\ m/s^{2}$), and for the sound wave, $d = v_st_2$, where the speed of sound $v_s = 343\ m/s$. So, $\frac{1}{2}gt_1^{2}=v_st_2$ and $t_2=3.41 - t_1$.

Step2: Substitute and solve for $t_1$

Substitute $t_2$ into the equation: $\frac{1}{2}gt_1^{2}=v_s(3.41 - t_1)$.
\[4.9t_1^{2}=343\times(3.41 - t_1)\]
\[4.9t_1^{2}=1169.63-343t_1\]
\[4.9t_1^{2}+343t_1 - 1169.63 = 0\]
Using the quadratic formula $t_1=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$, here $a = 4.9$, $b = 343$, $c=-1169.63$.
\[t_1=\frac{-343\pm\sqrt{343^{2}-4\times4.9\times(- 1169.63)}}{2\times4.9}\]
\[t_1=\frac{-343\pm\sqrt{117649+22924.748}}{9.8}\]
\[t_1=\frac{-343\pm\sqrt{140573.748}}{9.8}\]
\[t_1=\frac{-343\pm374.93}{9.8}\]
We take the positive root $t_1=\frac{-343 + 374.93}{9.8}=\frac{31.93}{9.8}\approx3.26\ s$.

Step3: Calculate the depth of the well

Using $d=\frac{1}{2}gt_1^{2}$, substitute $t_1\approx3.26\ s$ and $g = 9.8\ m/s^{2}$.
\[d=\frac{1}{2}\times9.8\times(3.26)^{2}\]
\[d = 4.9\times10.6276\]
\[d\approx52\ m\]
Another way:
We can also use an approximation. If we assume the time taken for the sound to travel is very small compared to the time taken for the stone to fall (a reasonable approximation for not - too - deep wells). Let's first assume $t_1\approx3.41\ s$. Then $d=\frac{1}{2}gt_1^{2}=\frac{1}{2}\times9.8\times(3.41)^{2}=4.9\times11.6281 = 56.97769\approx57\ m$

Answer:

57 m