An echo is returned in 3 s. What is the distance of the reflecting surface from the source, considering the speed of sound as 342 ms -1 ?
513 m
Understanding Echoes and Distance Calculation
This problem asks us to find the distance to a reflecting surface using the time it takes for an echo to return and the speed of sound in the medium. An echo is a sound wave that reflects off a surface and returns to the listener.
When we hear an echo, the sound has traveled from the source to the reflecting surface and then back from the reflecting surface to the source. The total time recorded for the echo is the time taken for this round trip.
Given Information
- Total time for the echo to return ($\Delta t$): 3 seconds
- Speed of sound ($v$): 342 meters per second
Calculating the One-Way Travel Time
Since the total time is for the sound to travel to the surface and return, the time taken for the sound to travel just one way (from the source to the reflecting surface) is half of the total time.
One-way time ($t_{\text{one-way}}$) = Total time / 2
$\Delta t_{\text{one-way}} = \frac{\Delta t}{2}$
$\Delta t_{\text{one-way}} = \frac{3 \text{ s}}{2} = 1.5 \text{ s}$
Applying the Distance Formula
The relationship between distance, speed, and time is given by the formula:
Distance = Speed × Time
To find the distance from the source to the reflecting surface, we use the speed of sound and the one-way travel time.
$d = v \times \Delta t_{\text{one-way}}$
Step-by-Step Calculation
- Identify the total time for the echo: 3 s.
- Identify the speed of sound: 342 m/s.
- Calculate the one-way time for the sound to reach the reflecting surface: $\Delta t_{\text{one-way}} = \frac{3 \text{ s}}{2} = 1.5 \text{ s}$.
- Use the distance formula $d = v \times \Delta t_{\text{one-way}}$.
- Substitute the values: $d = 342 \text{ m/s} \times 1.5 \text{ s}$.
- Calculate the distance: $d = 513 \text{ m}$.
Therefore, the distance of the reflecting surface from the source is 513 meters.
Result Summary
| Quantity | Symbol | Value | Units |
|---|---|---|---|
| Total Echo Time | $\Delta t$ | 3 | s |
| Speed of Sound | $v$ | 342 | m/s |
| One-Way Time | $\Delta t_{\text{one-way}}$ | 1.5 | s |
| Distance to Surface | $d$ | 513 | m |
Revision Table: Key Concepts for Echo Problems
| Concept | Explanation | Formula |
|---|---|---|
| Echo Definition | A reflected sound wave returning to the source. | N/A |
| Total Echo Time | Time taken for sound to travel to reflector and back. | $\Delta t_{\text{total}}$ |
| One-Way Time | Time taken for sound to travel from source to reflector. | $\Delta t_{\text{one-way}} = \frac{\Delta t_{\text{total}}}{2}$ |
| Distance Calculation | Distance = Speed × Time (using one-way time) | $d = v \times \Delta t_{\text{one-way}}$ |
Additional Information: Factors Affecting Speed of Sound
The speed of sound in air is not constant; it can be affected by several factors:
- Temperature: The speed of sound increases with increasing temperature.
- Humidity: Higher humidity slightly increases the speed of sound.
- Medium: Sound travels at different speeds in different materials (solids, liquids, gases). It generally travels fastest in solids and slowest in gases.
- Pressure: For an ideal gas, the speed of sound is independent of pressure, but real gases can show a slight dependence.
In this problem, the speed of sound (342 m/s) is given, so we use that value directly. This speed is typical for sound in dry air at around 20°C.
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