At a height of 5 meters above the surface of a freshwater lake, a sound pulse is generated. The echo from the bottom of the lake returns to the point of origin 0.140 s later. The air and water temperatures are 20°C. How deep is the lake?
To determine the depth of the lake, we can use the speed of sound in water and the time it takes for the echo to return.
Step 1: Convert air temperature to Kelvin
The air temperature is given as 20°C. To convert it to Kelvin, we add 273.
20°C + 273 = 293 K
Step 2: Calculate the speed of sound in air
The speed of sound in air can be calculated using the formula:
v_air = 331.4 m/s + (0.6 m/s/K) * temperature in Kelvin
For 20°C, the speed of sound in air is:
v_air = 331.4 m/s + (0.6 m/s/K) * 293 K
v_air ≈ 343.28 m/s
Step 3: Calculate the speed of sound in water
The speed of sound in freshwater can be calculated using the formula:
v_water = 1402 + (4.7 * temperature in Celsius) - (0.055 * temperature in Celsius^2)
For 20°C, the speed of sound in water is:
v_water = 1402 + (4.7 * 20) - (0.055 * 20^2)
v_water ≈ 1484 m/s
Step 4: Calculate the total time taken for the round trip of the sound pulse
The total time is the time taken for the sound pulse to reach the bottom and echo back to the origin:
total_time = 2 * echo_time
total_time = 2 * 0.140 s
total_time = 0.28 s
Step 5: Calculate the time taken for the sound to reach the bottom
The time taken for the sound pulse to reach the bottom of the lake can be calculated using the formula:
distance = speed * time
Since the pulse traveled from the origin to the bottom of the lake, the distance is twice the height of the lake:
distance = 5 m * 2 = 10 m
The time taken for the sound to reach the bottom can be calculated by rearranging the formula:
time = distance / speed
For water, the speed is v_water:
time = 10 m / 1484 m/s
time ≈ 0.00673 s
Step 6: Subtract the time taken for the sound to reach the bottom from the total time
To find the time taken for the echo to return, we subtract the time taken for the sound to reach the bottom from the total time:
echo_time = total_time - time
echo_time = 0.28 s - 0.00673 s
echo_time ≈ 0.27327 s
Step 7: Calculate the depth of the lake
Finally, we can calculate the depth of the lake by dividing the speed of sound in water by 2 and multiplying it by the echo time:
depth = (v_water / 2) * echo_time
depth = (1484 m/s / 2) * 0.27327 s
depth ≈ 201.9157 m
Therefore, the depth of the lake is approximately 201.92 meters.
To determine the depth of the lake, we can use the speed of sound in both air and water.
First, let's calculate the speed of sound in air at 20°C. The speed of sound in air can be approximated using the formula:
v_air = 331.4 + 0.6 * T,
where T is the temperature in degrees Celsius.
For T = 20°C, the speed of sound in air is:
v_air = 331.4 + 0.6 * 20 = 343.4 m/s.
Next, we need to calculate the speed of sound in water at 20°C. The speed of sound in water is approximately 1482 m/s at this temperature.
Now, let's use the information provided to find the time it takes for the sound pulse to travel down to the bottom of the lake and back. The total time can be calculated as twice the time it takes for the sound to travel from the surface to the bottom:
Total time = 2 * Time for sound to reach the bottom of the lake.
Given that the echo from the bottom of the lake returns to the point of origin 0.140 s later, the time for the sound to reach the bottom is half of this value:
Time for sound to reach the bottom = 0.140 s / 2 = 0.07 s.
Now, we can use the speed of sound in water to calculate the depth of the lake using the formula:
Depth = Speed of sound in water * Time for sound to reach the bottom.
Depth = 1482 m/s * 0.07 s = 103.74 m.
Therefore, the depth of the lake is approximately 103.74 meters.