Question

The speed of sound is \(330 \mathrm{~m} / \mathrm{s}\). If fireworks are \(1 \mathrm{~km}\) away, what is the length of the time delay between seeing the fireworks explode and hearing the sound?

Short answer

The time delay is 3 seconds.

Step-by-step solution

Convert Distance to Meters

First, convert the distance from kilometers to meters since the speed of sound is given in meters per second. We know that 1 kilometer is equal to 1000 meters. So, multiply the distance in kilometers by 1000 to get the distance in meters.

Calculate Time Delay

Use the formula for time which is time = distance / speed. Substitute the distance you found in the first step and the given speed of sound to find the time delay.

Report the Answer

After calculating the time delay in seconds, report the answer. Make sure the unit of time you provide is appropriate for the context, in this case, seconds.

Key concepts

Sound Propagation

Understanding how sound waves travel through various media is essential when calculating sound propagation. Sound travels at different speeds depending on the material through which it moves. In air, at a temperature of 20 degrees Celsius, the speed of sound is approximately 343 meters per second (m/s), but this value can vary slightly with temperature. Sound propagation involves not only the speed at which the sound waves travel but also the direction and manner in which they spread out from the source.

When we observe fireworks, the light reaches us almost instantaneously because the speed of light (about 299,792 kilometers per second in a vacuum) is vastly greater than the speed of sound. However, sound waves take a longer time to travel the same distance, leading to a noticeable delay between seeing and hearing an event. This delay is what we experience when watching fireworks from a distance. The science behind this is straightforward but requires an understanding of the basic principles of wave movement and speed.

Distance Conversion

In various calculations, especially when dealing with the speed of sound, it's crucial to have a consistent unit of measurement. When you receive a distance in kilometers but the speed of sound is given in meters per second, you need to convert the distance from kilometers to meters. This is an essential step because the units of measure for speed must match the units for distance in the calculation.

To convert kilometers to meters, remember that 1 kilometer is equivalent to 1,000 meters. So, you multiply the given distance by 1,000. For example, if fireworks are 1 km away, the distance in meters is 1 km x 1000 m/km, which equals 1000 meters. This conversion ensures that the units will be consistent when you calculate the time it takes for the sound to travel from the fireworks to your position.

Time Delay Calculation

After converting the distance into the proper units, the final step is to calculate the time delay. The formula to find the time it takes for sound to travel a certain distance is given by the equation \( time = \frac{distance}{speed} \), where 'time' is the time delay, 'distance' is the distance the sound needs to travel, and 'speed' is the speed of sound. Once you've substituted the values into the formula, you can solve for time.

Using the example of the fireworks being 1 km (or 1000 meters after conversion) away and the speed of sound at about 330 m/s, the time delay is calculated as \( time = \frac{1000 \, m}{330 \, m/s} \), which simplifies to approximately 3.03 seconds. The calculated time delay is the time duration between when you see the fireworks explode and when you hear the sound they make. Understanding how to calculate this time delay is useful, not only for solving textbook problems but also for real-world situations where sound travel time plays a crucial role.