In air at normal conditions, what is the minimum distance in metres between a sound source and a reflecting surface that is required to hear a distinct echo?

Introduction / Context:
An echo is a separate repetition of a sound due to reflection from a distant surface such as a wall, building, or hill. For the human ear to perceive an echo as distinct from the original sound, there must be a sufficient time gap between the original sound and the reflected sound. This time gap translates to a minimum distance for the reflecting surface, based on the speed of sound in air. This question asks you to recall or derive that minimum distance under normal conditions.

Given Data / Assumptions:

• The medium is air at ordinary temperature and pressure.
• The speed of sound in air is approximately 340 m/s (or 343 m/s; we use 340 m/s for easy calculation).
• The human ear can distinguish two sounds as separate if they reach the ear at least about 0.1 second apart.
• Sound must travel from the source to the reflecting surface and back to the listener.

Concept / Approach:
Let t be the minimum time interval required for the ear to perceive two sounds separately, and v be the speed of sound. For an echo, the sound travels a total distance of 2 * d, where d is the distance between the source and the reflecting surface. The total time taken is t = (2 * d) / v. Rearranging this gives d = (v * t) / 2. Substituting the typical values v approximately 340 m/s and t approximately 0.1 s gives a minimum distance of about 17 m.

Step-by-Step Solution:
Step 1: Use the relation time = distance / speed. For the echo path, total distance travelled by sound is 2 * d. Step 2: Write t = (2 * d) / v, where t is the time interval between original and reflected sound reaching the listener. Step 3: Rearrange for distance: d = (v * t) / 2. Step 4: Substitute v = 340 m/s and t = 0.1 s: d = (340 * 0.1) / 2. Step 5: Compute: 340 * 0.1 = 34; then 34 / 2 = 17 m. So the minimum distance needed is about 17 metres.

Verification / Alternative check:
If the reflecting surface were only 10 m away, the total distance for the sound to travel out and back would be 20 m. With v = 340 m/s, the time taken is 20 / 340, which is about 0.059 s, less than 0.1 s, so the reflected sound would blend with the original. At 17 m, the round trip distance is 34 m, giving a travel time of 34 / 340 = 0.1 s, exactly the minimum separation required. This confirms that the standard textbook value of 17 m is consistent with human hearing limits and speed of sound in air.

Why Other Options Are Wrong:
10 m: Too small; the time difference between original and reflected sound is less than 0.1 s, so no distinct echo is heard. 13 m: Also too small; round trip distance is 26 m, giving a time of about 0.076 s, still below the 0.1 s threshold. 21 m: Greater than the minimum required distance, so an echo would be heard, but the question asks for the minimum distance, which is smaller and equal to 17 m.

Common Pitfalls:
Students often forget that the sound travels to the wall and back, so they mistakenly use d instead of 2 * d in the time calculation. Others may use an approximate speed of sound but forget to include the ear resolution time of about 0.1 s. To avoid mistakes, always remember that for an echo you must consider the round trip path and the minimum time separation that the human ear can distinguish.

Final Answer:
The minimum distance required between the sound source and the reflecting surface to hear a distinct echo in air is 17 m.