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# A splash is heard after 3.12s of a stone is dropped into a well 45m deep. The speed of sound in air is: [$g = 10m{s^{ - 2}}$](A) $330m/s$ (B) $375m/s$(C) $340m/s$(D) $346m/s$

Last updated date: 13th Jun 2024
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Hint: We need to calculate the time taken by the stone to reach the well at first and deduct that amount from the total time, which is $3.12s$. Depth of the well is given, therefore, this remaining time was taken by sound to reach the observer, and hence we can find the

Complete step by step solution:
In the question it is given that the splash is heard after $3.12s$, this means that the stone reached the well and then the sound travelled back to the observer, this took $3.12s$.
The following values are given to us:
Time after which the splash is heard $= 3.12s$$(t)$
Depth of the well $= 45m$
g $= 10m{s^{ - 2}}$
Let ${t_1}$ be the time taken by the stone to reach the water.
And $h$ be the depth of the well $= 45m$
We know, the three equations of motion, from there let us consider the equation which helps us to calculate the distance travelled from initial velocity, acceleration and time taken.
$h = ut + \dfrac{1}{2}g{t_1}^2$
Since the stone is thrown, we consider the initial velocity to be zero.
$\therefore U = 0$
So, $h = \dfrac{1}{2}g{t_1}^2$
Now, rearranging the equation, we get:
${t_1} = \sqrt {\dfrac{{2h}}{g}}$
Putting the values, we obtain:
${t_1} = \sqrt {\dfrac{{2 \times 45}}{{10}}} s$
On solving, we get:
${t_1} = 3s$
Total time taken by the splash to reach the observer is $= 3.12s$, therefore, time taken for the sound to travel to the observer is: ${t_2} = t - {t_1}$
Therefore, ${t_2} = (3.12 - 3)s = 0.12s$
We know. Distance travelled by the sound is depth of the well, which is $= 45m$
Thus, we can calculate the speed of sound, as we know:
$\Rightarrow$ $Speed = \dfrac{{{\text{Distance travelled}}}}{{{\text{Time Taken}}}}$
Putting the values, we obtain:
$\Rightarrow$ $Speed = \dfrac{{45}}{{0.12}}$
Thus, we finally arrive at:
$\Rightarrow$ $Speed = 375m/s$
This is the required solution.

Therefore, option (B) is correct.

Note: SI unit of speed is $m/s$, so if we have distances given in kilometre and time given I hours, we must convert them to meters and seconds respectively, and not write $km/hr$, as the unit of speed. Initial velocity of the stone is taken to be zero, as the velocity remains the same, with respect to the reference frame.