A boy drops a 0.10 kg stone down a 150 m well and listens for the echo. The air temperature is 20°C. How long after the stone is dropped will the boy hear the echo?
The speed of sound through air at 20°C is v = 343 m/s.
Downward motion of the stone:
h =g•t1²/2, t1 =sqrt(2•h/g) = sqrt(2•150/9.8) = 5.5 s.
Upward motion of sound:
h = v•t2.
t2 = h/v = 150/343 =0.44s.
t =t1+t2 =5.5 +0.44 =5.94 s.
To find out how long after the stone is dropped the boy will hear the echo, we need to consider the time it takes for the sound to travel down to the bottom of the well and back up again.
First, let's calculate the time it takes for the stone to reach the bottom of the well:
Since we know the distance the stone falls (150 m) and the acceleration due to gravity (9.8 m/s^2), we can use the kinematic equation:
d = (1/2) * g * t^2
where:
d = distance
g = acceleration due to gravity
t = time
Rearranging the equation to solve for time:
t = √(2d / g)
Plugging in the values:
t = √(2 * 150 m / 9.8 m/s^2)
t ≈ 5.37 s
So, it takes approximately 5.37 seconds for the stone to reach the bottom of the well.
Next, we need to calculate the time it takes for the sound to travel back up the well:
Since we know the speed of sound through air (343 m/s) and the distance the sound has to travel (150 m), we can use the equation:
d = v * t
where:
d = distance
v = velocity (speed of sound)
t = time
Rearranging the equation to solve for time:
t = d / v
Plugging in the values:
t = 150 m / 343 m/s
t ≈ 0.437 s
So, it takes approximately 0.437 seconds for the sound to travel back up the well after the stone hits the bottom.
The total time it takes for the boy to hear the echo is the sum of the time it takes for the stone to reach the bottom and the time it takes for the sound to travel back up:
Total time = time for stone to reach bottom + time for sound to travel back up
Total time ≈ 5.37 s + 0.437 s
Total time ≈ 5.807 s
Therefore, the boy will hear the echo approximately 5.807 seconds after the stone is dropped.