Calculating Echo Time Interval

This problem involves calculating the time it takes for sound to travel to a wall and return as an echo. Understanding how echoes work is key here.

What is an Echo?

An echo is the reflection of sound waves from a surface back to the listener. For an echo to be heard, the sound must travel from the source, reflect off an obstacle, and return to the listener.

Analyzing the Problem

We are given the following information:

• Distance from the boy to the wall: $d = 83 \, \text{m}$
• Speed of sound in the medium: $v = 332 \, \text{m/s}$

The sound travels from the boy to the wall and then back to the boy. Therefore, the total distance covered by the sound wave to produce the echo is twice the distance to the wall.

Calculating Total Distance Travelled by Sound

The total distance ($D$) the sound travels for the echo is:

$\text{Total distance} = 2 \times \text{Distance to the wall}$

$D = 2 \times d$

$D = 2 \times 83 \, \text{m} = 166 \, \text{m}$

Using the Speed, Distance, Time Relationship

The relationship between speed, distance, and time is given by:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

We need to find the time interval ($t$), so we rearrange the formula:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

$t = \frac{D}{v}$

Calculating the Time Interval for the Echo

Now, substitute the values we have for the total distance and the speed of sound:

$t = \frac{166 \, \text{m}}{332 \, \text{m/s}}$

Let's perform the division:

$t = 0.5 \, \text{s}$

So, the time interval when the boy hears the echo is 0.5 seconds.

Summary of Calculation Steps

1. Identify the distance to the reflecting surface ($d$).
2. Identify the speed of sound ($v$).
3. Calculate the total distance the sound travels (to the wall and back), which is $2d$.
4. Use the formula $t = \frac{\text{Distance}}{\text{Speed}}$ to find the time interval.

Applying these steps:

• Distance to wall $d = 83 \, \text{m}$
• Speed of sound $v = 332 \, \text{m/s}$
• Total distance $D = 2 \times 83 \, \text{m} = 166 \, \text{m}$
• Time $t = \frac{166 \, \text{m}}{332 \, \text{m/s}} = 0.5 \, \text{s}$

The calculated time interval is 0.5 s.

Revision Table: Key Concepts for Echo Calculation

Additional Information: Factors Affecting Echoes

The time it takes to hear an echo can be affected by several factors:

• Distance: A greater distance to the reflecting surface results in a longer time delay for the echo.
• Speed of Sound: The speed of sound depends on the medium (like air, water, solids) and its temperature. Higher temperature in air generally increases the speed of sound, reducing the echo time for a fixed distance.
• Reflecting Surface: The surface must be large enough and hard enough to reflect sound effectively. Soft or irregular surfaces absorb sound rather than reflecting it.
• Minimum Distance: For a distinct echo to be heard by humans, the time delay between the original sound and the echo must be at least about 0.1 seconds (persistence of hearing). This corresponds to a minimum distance to the reflector, which depends on the speed of sound. For a speed of 340 m/s, this minimum distance is approximately $(340 \, \text{m/s} \times 0.1 \, \text{s}) / 2 = 17 \, \text{m}$. In our problem, the distance is 83m, which is much greater than the minimum distance required for hearing a distinct echo.