# Thunder and Lightning

Alignments to Content Standards: 7.RP.A

Alyssa sees a lightning bolt in the sky and counts four seconds until she hears the thunder.

1. There are 5280 feet in a mile and about 3.28 feet in a meter. Given that sound travels about 343 meters per second, is the lightning strike within one mile of Alyssa?
2. What is the speed of sound in miles per hour? <./li>

## IM Commentary

The purpose of this task is to work on performing unit conversions in a real world context about the speed of sound. While the nature of the task appears to fit 6.RP.3d, the numbers involved make this a 7th grade task (6th graders work primarily with ratios of whole numbers). Seeing lightning and waiting to hear the accompanying thunder is a common experience. This task examines one side of this scenario: the time it takes for the sound from the lightning strike to reach Alyssa. The speed of sound is most often given in meters per second in a scientific context but sometimes also in miles per hour. In this task, students work first with the scientific units and then convert to the more familiar miles per hour: this provides extra practice with unit conversions and also helps students to gain an intuition for how fast sound travels. In particular, sound travels about ten times as fast as a car on the freeway and about one and a half times as fast as a passenger airplane.

A short video of a lightning strike followed by the accompanying thunder can be seen here:

The teacher may wish to show the video first and prompt the students to explain the time delay in terms of the relative speeds of light and sound.

The speed of light, a little over 186,000 miles per second, has been left out of consideration in the problem statement and solution. Because light travels so fast, even if the lightning were 100 miles away it would be visible in less than one thousandth of a second: in all cases, this amount of time is smaller than the smallest significant digit for when the thunder is audible and so does not impact the answer. Since taking the speed of light into account complicates the problem without influencing the answer, it has been left out but the teacher may wish to discuss this if it comes up in student work.

Students working on this task will need to Attend to Precision (MP6), understanding that the conversion rate of 3.28 feet per meter is only approximate as is the 343 meters per second speed of sound. This means that students must maintain 3 significant digits in their calculations and the rounding should only take place after all calculations have been completed. Interestingly, the calculated value differs from the 768 miles per hour found on wikipedia https://en.wikipedia.org/wiki/Speed_of_sound because the numbers in 343 meters per second and 3.28 feet per meter are only approximate (with three significant digits in each case). This shows that dealing accurately with significant digits requires very close attention.

## Solution

1. Since the thunder from the lightning strike travels at about 343 meters per second, in 4 seconds the sound will have travelled about $$4 \text{ seconds} \times 343 \frac{\text{meters}}{\text{second}} = 1372 \text{ meters}.$$ Since there are about 3.28 feet in a meter this is $$1372 \text{ meters} \times 3.28 \frac{\text{feet}}{\text{meter}} \approx 4500 \text{ feet}.$$ This is less than a mile.

Alternatively, if Alyssa is familiar with track distances, she knows that a ''metric mile'' is 1600 meters and so 1372 meters will be less than a mile. The metric mile is a little bit shorter than a mile but since tracks are usually built with 400 meters in a lap, this is a convenient substitute for a mile and the two are very close (1600 meters is about 5249 feet).

2. To convert from meters per second to miles per hour we have to convert both the distances and the times. We will do these separately and then put all of the calculations together to find the speed of the thunder in miles per hour. First we convert 343 meters to miles:

\begin{align} 343 \text{ meters} &= 343 \text{ meters} \times 3.28 \frac{\text{feet}}{\text{meter}} \times \frac{1}{5280} \frac{\text{mile}}{ \text{feet}} \\ &= \frac{343 \times 3.28}{5280} \text{ miles} \\ &\approx 0.213 \text{ miles}. \end{align}

Next we convert seconds to hours:

\begin{align} 1 \text{ second} &= 1 \text{ second} \times \frac{1}{60} \frac{\text{minute}}{\text{seconds}} \times \frac{1}{60} \frac{\text{hour}}{ \text{minutes}} \\ &= \frac{1}{3600} \text{hours}. \end{align}

Equivalently, we have 1 hour = 3600 seconds. When we combine all of our unit conversions, we use the complete expression $\frac{343 \times 3.28}{5280}$ miles instead of the rounded $0.213$ miles since the rounding should take place after all operations have been performed. Using our calculations we find

\begin{align} 343 \frac{\text{meters}}{\text{second}} &\approx \frac{340 \times 3.28}{5280} \frac{\text{miles}}{\text{second}} \\ &= \frac{343 \times 3.28}{5280} \frac{\text{miles}}{\text{second}} \times 3600 \frac{\text{seconds}}{\text{hour}} \\ &\approx 767 \frac{\text{miles}}{\text{hour}} \end{align}