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Chapter 2: Problem 12

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.)

Short Answer

Expert verified
The lightning strike was approximately 1.6 miles away.

Step by step solution

01

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.
02

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.
03

Calculate the Distance in Meters

The distance \(d\) can be calculated using the formula: \[d = ext{{speed}} \times ext{{time}} = 344 \mathrm{~m/s} \times 7.5 \mathrm{~s} = 2580 \mathrm{~m}\]
04

Convert the Distance from Meters to Miles

To convert meters to miles, use the conversion factor \(1 \mathrm{~mile} = 1609.34 \mathrm{~meters}\). Thus, \[\text{Distance in miles} = \frac{2580 \mathrm{~m}}{1609.34 \mathrm{~m/mile}} \approx 1.6 \text{ miles} \]
05

Review Your Calculation

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(v\) 脳 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.

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Most popular questions from this chapter

According to recent typical test data, a Ford Focus travels \(0.250 \mathrm{mi}\) in \(19.9 \mathrm{~s},\) starting from rest. The same car, when braking from \(60.0 \mathrm{mi} / \mathrm{h}\) on dry pavement, stops in \(146 \mathrm{ft}\). Assume constant acceleration in each part of its motion, but not necessarily the same acceleration when slowing down as when speeding up. (a) Find this car's acceleration while braking and while speeding up. (b) If its acceleration is constant while speeding up, how fast (in \(\mathrm{mi} / \mathrm{h}\) ) will the car be traveling after \(0.250 \mathrm{mi}\) of acceleration? (c) How long does it take the car to stop while braking from \(60.0 \mathrm{mi} / \mathrm{h} ?\)

Two coconuts fall freely from rest at the same time, one from a tree twice as high as the other. (a) If the coconut from the taller tree reaches the ground with a speed \(V,\) what will be the speed (in terms of \(V\) ) of the coconut from the other tree when it reaches the ground? (b) If the coconut from the shorter tree takes time \(T\) to reach the ground, how long (in terms of \(T\) ) will it take the other coconut to reach the ground?

A mouse travels along a straight line; its distance \(x\) from the origin at any time \(t\) is given by the equation \(x=\) \(\left(8.5 \mathrm{~cm} \cdot \mathrm{s}^{-1}\right) t-\left(2.5 \mathrm{~cm} \cdot \mathrm{s}^{-2}\right) t^{2} .\) Find the average velocity of the mouse in the interval from \(t=0\) to \(t=1.0 \mathrm{~s}\) and in the interval from \(t=0\) to \(t=4.0 \mathrm{~s}\)

If the aorta (diameter \(d_{\mathrm{a}}\) ) branches into two equal-sized arteries with a combined area equal to that of the aorta, what is the diameter of one of the branches? A. \(V d_{\text {a }}\) B. \(d_{\text {a }} / \sqrt{2}\) C. \(2 d_{\mathrm{a}}\) D. \(d_{\mathrm{a}} / 2\)

A car in the northbound lane is sitting at a red light. At the moment the light turns green, the car accelerates from rest at \(2 \mathrm{~m} / \mathrm{s}^{2}\). At this moment, there is also a car in the southbound lane that is \(200 \mathrm{~m}\) away and traveling at a constant \(25 \mathrm{~m} / \mathrm{s}\). The northbound car maintains its acceleration until the two cars pass each other. (a) How long after the light turns green do the cars pass each other? (b) How far from the red light are they when they pass each other?
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