How much faster does sound travel on a hot summer day, 40 °C, than on the coldest winter day, -40 °C?
To determine how much faster sound travels on a hot summer day compared to the coldest winter day, we need to understand the relationship between temperature and sound speed.
The speed of sound is directly proportional to the square root of the temperature. This relationship is described by the formula:
v = √(γ * R * T)
Where:
v = Speed of sound
γ = Adiabatic index or heat capacity ratio of the gas (approximately 1.4 for air)
R = Gas constant for air (approximately 287 J/(kg*K))
T = Temperature in Kelvin (K)
First, let's convert the given temperatures into Kelvin by adding 273.15 to each value:
Hot summer day temperature = 40 °C + 273.15 = 313.15 K
Coldest winter day temperature = -40 °C + 273.15 = 233.15 K
Next, we can calculate the speed of sound at each temperature. Plugging in the values into the formula:
For the hot summer day:
v_hot = √(1.4 * 287 * 313.15)
For the coldest winter day:
v_cold = √(1.4 * 287 * 233.15)
Using a calculator, we can evaluate these expressions:
v_hot ≈ 351.46 m/s
v_cold ≈ 301.85 m/s
Therefore, sound travels approximately 351.46 m/s on a hot summer day (40 °C) and 301.85 m/s on the coldest winter day (-40 °C). The difference in speed is approximately 49.61 m/s (rounded to two decimal places).