An object is 10.64 km below the sea level. A research team sends down a sonar signal to confirm this depth. After how long can it expect to get the echo? Take speed of sound in sea water = 1520 m/s.

Question

Prepp Team · Accepted Answer

Calculating Sonar Echo Time Below Sea Level

This question asks us to calculate the time it takes for a sonar signal to travel to an object 10.64 km below sea level and for the echo to return. Sonar uses sound waves to determine depth or locate objects underwater. The signal travels from the source, reflects off the object, and returns to the source as an echo.

Understanding the Sonar Principle and Distance

When a sonar signal is sent down, it travels to the object and then the echo travels back up to the research team. This means the sound wave covers the distance from the source (sea surface) to the object and back again. Therefore, the total distance traveled by the sonar signal is twice the depth of the object.

Depth of the object below sea level = 10.64 km
Speed of sound in sea water = 1520 m/s

Step-by-Step Calculation of Echo Time

First, we need to ensure all units are consistent. The depth is given in kilometers (km), and the speed is given in meters per second (m/s). We should convert the depth from kilometers to meters.

Conversion:

1 km = 1000 meters

Depth in meters $= 10.64 \, \text{km} \times 1000 \, \text{m/km}$

Depth in meters $= 10640 \, \text{m}$

Next, we determine the total distance the sonar signal travels.

Total distance = Distance down + Distance up

Since the distance down is the depth and the distance up is also the depth (assuming a flat reflection), the total distance is:

Total distance $= 2 \times \text{Depth}$

Total distance $= 2 \times 10640 \, \text{m}$

Total distance $= 21280 \, \text{m}$

Now we can use the relationship between distance, speed, and time:

Distance = Speed $\times$ Time

We want to find the time, so we rearrange the formula:

Time = Distance / Speed

Substitute the values we have:

Time $= \frac{21280 \, \text{m}}{1520 \, \text{m/s}}$

Let's perform the division:

Time $= 14 \, \text{s}$

So, the research team can expect to get the echo after 14 seconds.

Here is a summary of the calculation steps:

Parameter	Value	Unit
Object Depth	10.64	km
Object Depth (converted)	10640	m
Speed of Sound	1520	m/s
Total Distance (down & back)	21280	m
Expected Echo Time	14	s

Conclusion

Based on the depth of 10.64 km and the speed of sound in seawater at 1520 m/s, the sonar signal travels a total distance of 21280 m (down and back up). Using the formula Time = Distance / Speed, the calculated echo time is 14 seconds.

Revision Table: Key Concepts for Sonar Echo Time

Concept	Explanation	Formula/Relation
Sonar Principle	Uses sound waves to detect objects and measure distances underwater. Based on sending a pulse and receiving an echo.	N/A
Echo Time	The total time taken for the sound pulse to travel to the object and for the echo to return.	Time = Distance / Speed
Total Distance in Sonar	The distance covered by the sound wave is twice the depth or distance to the object, as it travels down and back up.	Total Distance = $2 \times \text{Depth}$
Units Consistency	It is crucial to use consistent units (e.g., meters for distance, seconds for time, meters/second for speed) in calculations.	1 km = 1000 m

Additional Information: Sonar Applications and Factors Affecting Speed of Sound

Sonar technology is widely used for various purposes underwater:

Navigation: Determining depth (bathymetry) to create maps of the seafloor.
Fishing: Locating fish schools.
Scientific Research: Studying marine life, geological formations, and ocean currents.
Underwater Communication: Sending messages acoustically.
Military Applications: Submarine detection, mine hunting, navigation.

The speed of sound in water, especially seawater, is not constant. It is primarily affected by three factors:

Temperature: Higher temperature generally increases the speed of sound.
Salinity: Higher salinity (saltiness) generally increases the speed of sound.
Pressure: Higher pressure (due to increasing depth) generally increases the speed of sound.

In real-world sonar applications, the speed of sound might vary with depth, requiring more complex calculations. However, in this problem, a constant speed of sound is provided for simplicity.