To determine the depth of the well, we can use the equation of motion for free fall. The stone falls from rest, so its initial velocity (u) is 0 m/s. The acceleration due to gravity (g) is -9.8 m/s² (negative since it acts in the opposite direction of the stone's motion).
Let's denote the time taken for the stone to reach the bottom of the well as t. The time taken for the sound to travel back up the well is 2.02 seconds. Since the sound travels at a velocity of 341 m/s, the distance it covers is 341 * 2.02 = 689.82 meters.
Using the equation of motion:
s = ut + (1/2)gt²
Here, s represents the distance traveled by the stone (the depth of the well). Plugging in the values:
s = 0 * t + (1/2) * (-9.8) * t² s = -4.9t²
Now, we know that the time taken for the sound to travel back up the well is t + 2.02 seconds. So the equation becomes:
689.82 = -4.9(t + 2.02)²
Solving this equation for t will give us the time it took for the stone to fall to the bottom of the well. Then we can substitute this value back into the equation to find the depth (s) of the well.
689.82 = -4.9(t² + 4.08t + 4.0804) 689.82 = -4.9t² - 20.052t - 20.052
Rearranging the equation:
4.9t² + 20.052t + 709.872 = 0
We can solve this quadratic equation to find the value of t. Using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
Here, a = 4.9, b = 20.052, and c = 709.872. Substituting these values into the formula and solving for t, we find two possible values for t: t = 6.127 or t = -28.997.
Since time cannot be negative in this context, we discard the negative value. Therefore, t = 6.127 seconds.
Finally, we substitute the value of t into the equation for s:
s = -4.9 * (6.127)² s = -4.9 * 37.511 s = -183.7609
Since the depth of the well cannot be negative, we take the magnitude of the calculated value:
Depth of the well = |s| = 183.7609 meters.
Therefore, the depth of the well is approximately 183.76 meters.