An ultrasonic ruler, such as the one discussed in Example 4 in Section 16.6, displays the distance between the ruler and an object, such as a wall. The ruler sends out a pulse of ultrasonic sound and measures the time it takes for the pulse to reflect from the object and return. The ruler uses this time, along with a preset value for the speed of sound in air, to determine the distance. Suppose that you use this ruler under water, rather than in air. The actual distance from the ultrasonic ruler to an object is 24.0 m. The adiabatic bulk modulus and density of seawater are Bad = 2.37 × 109 Pa and rho = 1025 kg/m3, respectively. Assume that the ruler uses a preset value of 343 m/s for the speed of sound in air. Determine the distance reading that the ruler displays.

reading would be 5.413 m.

Explanation:

Given:-

- The actual distance from ruler to an object is d = 24.0 m

- The adiabatic bulk modulus, B = 2.37 *10^9 Pa

- The density of seawater, ρ = 1025 kg/m^3

- The preset value of speed of sound in air, v_th = 343 m/s.

Find:-

Determine the distance reading that the ruler displays.

Solution:-

- We will first determine the actual speed of the sound ( v_a) in sea-water which can be determined from the following formula:

                            v_a = √ (B /  ρ )

- Plug in the values in the relationship above and compute v_a:

                            v_a = √ ( 2.37 *10^9 /  1025  )        

                            v_a = 1520.59038 m/s

- The time taken (t) for for the sound to travel from source(ruler) to an object which is d distance away.

                            d = v_a*t

                            t = d / v_a

                            t = 24.0 / 1520.59038

                            t = 0.01578 s

- The distance reading on the ruler would be preset speed (v_th) of sound in air multiplied by the time taken(t).

                           reading = v_th*t

                           reading = (343)*(0.01578)

                                         = 5.413 m