Problem 12 - College Physics

Question

At room temperature, sound travels at a speed of about \(344 \mathrm{~m} / \mathrm{s}\) in air. You see a distant flash of lightning and hear the thunder arrive \(7.5 \mathrm{~s}\) later. How many miles away was the lightning strike? (Assume the light takes essentially no time to reach you.)

Step-by-Step Solution

Answer

The lightning strike was approximately 1.6 miles away.

Step 1: Understand the Problem

You need to find the distance of a lightning strike in miles, knowing the speed of sound and the time delay between seeing the lightning and hearing the thunder.

Step 2: Know the Given Values and Units

The speed of sound is given as \(344 \mathrm{~m/s}\) and the time delay is \(7.5 \mathrm{~s}\). We are asked to find the distance in miles.

Step 3: Calculate the Distance in Meters

The distance \(d\) can be calculated using the formula: \[d = ext{{speed}} \times ext{{time}} = 344 \mathrm{~m/s} \times 7.5 \mathrm{~s} = 2580 \mathrm{~m}\]

Step 4: Convert the Distance from Meters to Miles

To convert meters to miles, use the conversion factor \(1 \mathrm{~mile} = 1609.34 \mathrm{~meters}\). Thus, \[\text{Distance in miles} = \frac{2580 \mathrm{~m}}{1609.34 \mathrm{~m/mile}} \approx 1.6 \text{ miles} \]

Step 5: Review Your Calculation

Check each step to ensure calculations were performed correctly, including the conversion from meters to miles.

Key Concepts

Distance Calculation

Calculating distance is all about determining how far an object travels over a certain amount of time, given its speed. In this exercise, we calculated the distance the sound traveled after a lightning strike. To find this, we used the fundamental formula for distance:

• Distance \(d\) = Speed \(v\) × Time \(t\)

In our problem, the speed of sound is given as \(344 \text{ m/s}\), and the time taken by the thunder to reach us is \(7.5\) seconds. Thus, plugging these values into the formula, we calculated:\[d = 344 \text{ m/s} \times 7.5 \text{ s} = 2580 \text{ meters}\]This calculation gives us the distance between the observer and the point of the lightning strike in meters. Calculating such distances can help understand how sound travels through different mediums.

Unit Conversion

Unit conversion is a crucial mathematical concept that allows us to express a quantity in different units. This is especially important in science and engineering, where measurements are often not in the desired unit.In this exercise, the purpose of unit conversion was to change the distance from meters to miles. The conversion factor for meters to miles is:

• 1 mile = 1609.34 meters

We use this conversion factor like a fraction, ensuring the units we don't need (meters) cancel out:\[\text{Distance in miles} = \frac{2580 \text{ meters}}{1609.34 \text{ meters/mile}}\]Calculating gives approximately \(1.6\) miles. This means the sound traveled every meter increment until it covered miles translating the given distance into a standard unit more commonly used for greater distances.

Sound Wave Propagation

Sound wave propagation describes how sound travels through different mediums, like air, water, or solids. The speed of sound varies based on the medium due to differences in density and elasticity. In air at room temperature, the speed of sound is approximately \(344 \text{ m/s}\).When lightning strikes, the sound (thunder) and light (flash) are produced simultaneously. However, they travel at vastly different speeds:

• Light travels almost instantaneously to the observer.
• Sound travels slower, requiring time to reach your ears.

Understanding sound wave propagation helps in estimating distances over which sound travels by knowing time delays between visual and auditory stimuli. Knowing how sound behaves as waves spreading out from a source helps in solving real-world problems and applications involving distances and speeds across various mediums.