Question

The minimum distance between source and reflecting surface for echo is :

• A. \(10.2 \ \text{m}\)
• B. \(17.2 \ \text{m}\)
• C. \(20.4 \ \text{m}\)
• D. \(27.4 \ \text{m}\)

Hint:

Key factors for echo calculation: - {Persistence of hearing:} For humans, this is about \(0.1 \ \text{s}\). This is the minimum time gap needed to distinguish the original sound from its echo. - {Speed of sound (\(v\)):} This depends on the medium and its temperature. For air at \(\approx 22^\circ\text{C}\), \(v \approx 344 \ \text{m/s}\). - {Formula:} The total distance sound travels is \(2d\). So, \(2d = v \times t\), which gives \(d = \frac{v \times t}{2}\). Always check if the problem provides a specific speed of sound; otherwise, use a standard room temperature value.

Answer 1

The Correct Option is B

Concept: An echo is the repetition of sound caused by the reflection of sound waves from a surface. For an echo to be heard distinctly, there must be a minimum time interval between the original sound and the reflected sound. This minimum time interval is known as the persistence of hearing. Step 1: Persistence of Hearing The human ear can distinguish between two sounds if the time interval between them is at least about \(0.1\) seconds. This is called the persistence of hearing. So, for an echo to be heard distinctly, the reflected sound must reach the ear at least \(0.1\) seconds after the original sound is produced. Let \(t = 0.1 \ \text{s}\). Step 2: Speed of Sound The speed of sound in air varies with temperature. At a room temperature of around \(20^\circ\text{C}\) to \(22^\circ\text{C}\), the speed of sound is approximately \(344 \ \text{m/s}\). We will use this value for calculation, as it directly leads to one of the options. Let \(v = 344 \ \text{m/s}\). Step 3: Calculating the Minimum Distance Let \(d\) be the minimum distance between the source of sound and the reflecting surface. For an echo, the sound travels from the source to the reflecting surface (distance \(d\)) and then travels back from the reflecting surface to the source/listener (another distance \(d\)). So, the total distance travelled by the sound for the echo to be heard is \(2d\). We know the basic relationship: distance = speed × time. Applying this to our echo scenario: \[ \text{Total distance} = \text{Speed of sound} \times \text{Time interval} \] \[ 2d = v \times t \] To find the minimum distance \(d\) to the reflector, we can rearrange the formula: \[ d = \frac{v \times t}{2} \] Step 4: Substitution and Calculation Substitute the values of \(v = 344 \ \text{m/s}\) and \(t = 0.1 \ \text{s}\) into the formula: \[ d = \frac{(344 \ \text{m/s}) \times (0.1 \ \text{s})}{2} \] \[ d = \frac{34.4 \ \text{m}}{2} \] \[ d = 17.2 \ \text{m} \] Thus, the minimum distance between the source and the reflecting surface for an echo to be heard distinctly is \(17.2 \ \text{m}\).