Math Problem Statement

A sonar echo returns to a submarine 0.6 s after being emitted. What is the distance to the object creating the echo? (Assume that the submarine is in the ocean, and the speed of sound in ocean water is 1540 m/s).

Express the numeral of your answer in SI units with 3 or more significant figures.

Solution

To calculate the distance to the object, we use the formula:

Distance to the object=Total distance traveled by sound2\text{Distance to the object} = \frac{\text{Total distance traveled by sound}}{2}

The total distance traveled by the sound is given by:

Total distance=speed of sound×time\text{Total distance} = \text{speed of sound} \times \text{time}

Given:

  • Speed of sound, v=1540m/sv = 1540 \, \text{m/s}
  • Time for the echo to return, t=0.6st = 0.6 \, \text{s}

Step 1: Calculate the total distance traveled by sound

Total distance=1540m/s×0.6s=924m\text{Total distance} = 1540 \, \text{m/s} \times 0.6 \, \text{s} = 924 \, \text{m}

Step 2: Calculate the one-way distance

Since the sound wave travels to the object and back, we divide the total distance by 2: Distance to the object=9242=462m\text{Distance to the object} = \frac{924}{2} = 462 \, \text{m}

Final Answer:

The distance to the object is 462 m.


Would you like a detailed explanation or further assistance?

Related Questions:

  1. How is the speed of sound affected by water temperature?
  2. What factors can alter the accuracy of sonar measurements in real-world scenarios?
  3. How does sonar differ from radar in terms of propagation speed?
  4. What applications rely on sonar technology for distance measurement?
  5. How do salinity and pressure influence the speed of sound in seawater?

Tip:

Always account for environmental variables like temperature, salinity, and pressure, as they significantly affect sound speed in water.

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Math Problem Analysis

Mathematical Concepts

Distance calculation
Sound wave physics
Basic arithmetic operations

Formulas

Distance = Speed × Time
Distance to object = Total distance traveled by sound ÷ 2

Theorems

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Suitable Grade Level

Grades 6-8