What is the speed of sound in air (a) on a cold winter day in Minnesota when the temperature is -25°F, and (b) on a hot summer day in Death Valley when the temperature is 125°F?

검증된 단계별 안내

Step 1: Convert the given temperatures from Fahrenheit to Celsius using the formula: \( T_{C} = \frac{5}{9}(T_{F} - 32) \). For part (a), substitute \( T_{F} = -25 \) into the formula. For part (b), substitute \( T_{F} = 125 \) into the formula.

Step 2: Use the formula for the speed of sound in air: \( v = 331.4 + 0.6 \cdot T_{C} \), where \( T_{C} \) is the temperature in Celsius. Substitute the converted temperatures from Step 1 into this formula for both cases.

Step 3: Simplify the expression for \( v \) for each case by performing the arithmetic operations. This will give the speed of sound in air for the cold winter day and the hot summer day.

Step 4: Ensure the units of the final speed are in meters per second (m/s), as the formula \( v = 331.4 + 0.6 \cdot T_{C} \) inherently provides the speed in m/s.

Step 5: Interpret the results conceptually: The speed of sound increases with temperature because warmer air molecules move faster, facilitating quicker propagation of sound waves.

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주요 개념

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Speed of Sound

The speed of sound is the distance that sound waves travel through a medium in a given amount of time. In air, this speed is influenced by factors such as temperature, humidity, and pressure. Generally, sound travels faster in warmer air because the molecules are more energetic and can transmit the sound waves more quickly.

Effect of Temperature on Sound Speed

Temperature significantly affects the speed of sound in air. As temperature increases, the speed of sound also increases due to the greater kinetic energy of air molecules, which facilitates faster propagation of sound waves. Conversely, at lower temperatures, the speed of sound decreases as molecular motion slows down.

Conversion of Temperature Units

To accurately calculate the speed of sound based on temperature, it is often necessary to convert temperatures from Fahrenheit to Celsius or Kelvin. The formula for conversion from Fahrenheit to Celsius is C = (F - 32) × 5/9. Understanding this conversion is essential for applying the correct temperature values in the speed of sound formula.