Understanding Sound Wave Travel Time
This problem asks us to determine the time it takes for a sound wave to travel a specific distance, given its frequency and wavelength. To solve this, we first need to find the speed of the sound wave, and then use the relationship between speed, distance, and time.
Given Information about the Sound Wave
- Frequency ($f$) = 1 kHz
- Wavelength ($\lambda$) = 50 cm
- Distance ($d$) = 1 km
Unit Conversion
Before we use the values in calculations, it's essential to convert them into standard SI units:
- Frequency: 1 kHz = $1 \times 1000$ Hz = 1000 Hz
- Wavelength: 50 cm = $50 / 100$ m = 0.5 m
- Distance: 1 km = $1 \times 1000$ m = 1000 m
Calculating the Speed of the Sound Wave
The speed ($v$) of a wave is given by the product of its frequency ($f$) and wavelength ($\lambda$). The formula is:
\(v = f \times \lambda\)
Let's plug in the converted values:
\(v = 1000 \text{ Hz} \times 0.5 \text{ m}\)
\(v = 500 \text{ m/s}\)
So, the speed of this sound wave is 500 meters per second.
Calculating the Time Taken to Travel the Distance
Now that we know the speed of the sound wave and the distance it needs to travel, we can calculate the time taken. The relationship between speed, distance, and time is:
\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)
We need to find the time, so we can rearrange the formula:
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
Using the given distance (in meters) and the calculated speed (in m/s):
\(t = \frac{1000 \text{ m}}{500 \text{ m/s}}\)
\(t = 2 \text{ s}\)
Therefore, it will take 2 seconds for the sound wave to travel 1 km.
Final Answer Determination
Based on our calculations, the time taken for the sound wave to travel 1 km is 2 seconds. This matches one of the provided options.
| Quantity | Value (Given) | Value (SI Units) |
|---|---|---|
| Frequency ($f$) | 1 kHz | 1000 Hz |
| Wavelength ($\lambda$) | 50 cm | 0.5 m |
| Distance ($d$) | 1 km | 1000 m |
| Speed ($v$) | - | 500 m/s (Calculated) |
| Time ($t$) | - | 2 s (Calculated) |
Revision Table: Wave Properties and Calculations
| Concept | Formula | Description |
|---|---|---|
| Wave Speed ($v$) | \(v = f \times \lambda\) | Relates the speed of a wave to its frequency and wavelength. |
| Time ($t$) | \(t = \frac{d}{v}\) | Relates the time taken to travel a distance ($d$) at a constant speed ($v$). |
| Frequency ($f$) | \(f = \frac{1}{T}\) | Number of wave cycles per second. (T is time period) |
| Wavelength ($\lambda$) | Distance between two consecutive points in phase on a wave. |
Additional Information: Sound Waves and Their Properties
Sound waves are mechanical waves that travel through a medium such as air, water, or solids. They are longitudinal waves, meaning the particles of the medium vibrate parallel to the direction of wave propagation.
- Frequency: Determines the pitch of the sound. Higher frequency means higher pitch. Measured in Hertz (Hz).
- Wavelength: The spatial period of the wave. Related to frequency and speed.
- Speed of Sound: Depends on the properties of the medium (e.g., temperature, density, elasticity). The speed of sound in air at room temperature is approximately 343 m/s, but in this specific problem, the wave properties define its speed as 500 m/s, which might be in a different medium or under different conditions.
- Amplitude: Determines the loudness or intensity of the sound.
- Time Period (T): The time taken for one complete wave cycle. \(T = \frac{1}{f}\).
Understanding these basic properties and their relationships is crucial for solving problems involving sound waves and other types of waves.
Important Questions from Traveling Sound Waves
Which one of the following statements about the speed of sound waves is not correct?
The amplitude of sound waves is measured in the units of
Which of the following statements is NOT correct regarding the travel of sound waves?
Which among the following is true for propagation of sound waves?
Which one of the following does not apply to sound waves in fluids?
