Time Calculation for Hearing a Splash: A 10m Well Example

The phenomenon of hearing a splash in a well demonstrates the interplay between the gravitational fall of an object and the propagation of sound through air. In this article, we will explore how to calculate the total time taken to hear the splash of a stone dropped into a 10-meter deep well, including both the fall time of the stone and the sound travel time.

Understanding the Problem

The problem involves two key components:

Time taken for the stone to reach the bottom of the well (t1): The stone is in free fall with an initial velocity of 0 m/s, and it travels a distance of 10 meters. Time taken for the sound to travel back up to the top of the well (t2): Sound travels through the air at a constant speed.

Step 1: Calculating the Fall Time of the Stone (t1)

When an object falls freely from rest, it can be described by the following free-fall formula:

d 1/2 g t12

Where:

#8226; d 10 meters (depth of the well) #8226; g 9.81 m/s2 (acceleration due to gravity) #8226; t1 time taken to fall

Rearranging the formula to solve for t12:

t122dg

Substituting the values:

t12×10text{m}9.81text{m}/2#8239;#8242;#8239;approx#8239;2.038.43text{ s}

Step 2: Calculating the Sound Travel Time (t2)

Sound travels through air at a constant velocity, which can be described by the formula:

t2 d/v

Where:

#8226; d 10 meters (depth of the well) #8226; v 344 m/s (speed of sound in air)

Substituting the values:

t210text{m}344text{m}/s#8239;#8242;#8239;approx#8239;0.029text{ s}

Calculating the Total Time (T)

The total time taken to hear the splash is the sum of the fall time (t1) and the sound travel time (t2):

Tt1 t2

Therefore:

T#8239;#8239;1.43text{ s} 0.029text{ s}#8239;#8242;#8239;approx#8239;1.459text{ s}

Conclusion

Thus, the total time taken to hear the splash of sound after the stone is dropped into the well is approximately 1.46 seconds.

Note: In practice, no splash is heard due to the dry well and potential deafness of the observers. However, for theoretical purposes, this calculation provides a clear insight into the relevant physics.