Question

Show the steps to convert the speed of sound, 344 meters per second, into miles per hour. [Section 1.6]

Short answer

To convert the speed of sound from meters per second (344 m/s) to miles per hour, first convert it to kilometers per hour by multiplying by the conversion factor 3.6: \(344 \times 3.6 = 1238.4 \textrm{ km/h}\). Then, convert this to miles per hour by multiplying by the conversion factor \(\frac{1}{1.60934}\): \(1238.4 \times \frac{1}{1.60934} \approx 769.3 \textrm{ mph}\). Therefore, the speed of sound is approximately 769.3 miles per hour.

Step-by-step solution

Convert meters per second to kilometers per hour

To convert meters per second to kilometers per hour, we need to multiply the speed by a conversion factor. There are 1000 meters in a kilometer and 3600 seconds in an hour, so the conversion factor is \(\frac{3600}{1000}=3.6\). Multiply the speed of sound in meters per second (344 m/s) by the conversion factor (3.6): 344 x 3.6 = 1238.4 kilometers per hour.

Convert kilometers per hour to miles per hour

Now we need to convert kilometers per hour to miles per hour. Since 1 mile is equal to approximately 1.60934 kilometers, the conversion factor is \(\frac{1}{1.60934}\). Multiply the speed of sound in kilometers per hour (1238.4 km/h) by the conversion factor (\(\frac{1}{1.60934}\)): 1238.4 x \(\frac{1}{1.60934}\) = 769.3 miles per hour (rounded to one decimal place). Thus, the speed of sound in miles per hour is approximately 769.3 mph.

Key concepts

Unit Conversion

Unit conversion is a crucial process in various fields of science, engineering, and everyday life. It allows us to translate a quantity expressed in one set of units to another, enabling comparison and computation across different systems of measurement. When we think about the speed of sound, it's often measured in meters per second (m/s). However, this may not be very intuitive for those more familiar with miles per hour (mph).

To convert from one unit to another, it's essential to know the correct conversion factors. As seen in the provided exercise, we first convert the speed from meters per second to kilometers per hour before changing into miles per hour. This two-step process simplifies the conversion and minimizes the potential for errors. Students should give special attention to using precise conversion factors, as they ensure the accuracy of the result. It’s also good practice to keep track of units at each step to avoid mixing them up.

Meters to Miles

Understanding how to convert meters into miles is a common and sometimes necessary calculation, especially when dealing with speeds or distances in different countries that use different units of measurement. One meter equals approximately 0.000621371 miles, but for many, calculating this in their heads is not feasible.

The textbook solution shows a conversion from meters to kilometers and then from kilometers to miles, utilizing the fact that 1 kilometer is roughly equivalent to 0.621371 miles. This conversion factor is a simplified version of the exact value (1 mile = 1.60934 kilometers). Once again, we see the importance of conversion factors. When teaching this concept, it's beneficial to explain why such conversions are necessary and to emphasize carrying out the process step-by-step to maintain accuracy.

Speed Measurement

Speed measurement is essential in understanding motion. It provides a way to describe how fast something is moving, which can be crucial information in many contexts, from daily commutes to scientific research. Speed is typically expressed as a distance traveled per unit of time, and common units include meters per second (m/s) and miles per hour (mph).

In science classes, students may encounter speed quantified in units that they are not familiar with, especially if they are in a country where the standard units differ from the metric system. By practicing conversions like in our exercise—which converts the speed of sound from the metric unit of meters per second to the imperial unit of miles per hour—students gain a more practical understanding of speed. Additionally, it helps to contextualize the meaning of speed in various settings, such as understanding velocity in aviation or the pace of a car journey.