Find the temperature at which the velocity of sound in air is 1.25 times the velocity of sound at 0 degree centigrade.
v = 331 m/s • sqrt (1 + T/273)
1.25•331 =331• sqrt (1 + T/273)
1+T/273 = 1.25²=1.56
T/273 = 1.56-1=0.56,
T=273•0.56 =153ºC
To find the temperature at which the velocity of sound in air is 1.25 times the velocity of sound at 0 degrees Celsius, we need to use the relationship between the velocity of sound in air and temperature.
The velocity of sound in air can be calculated using the formula:
v = sqrt(gamma * R * T)
Where:
v = velocity of sound
gamma = adiabatic index (for air, gamma is approximately 1.4)
R = specific gas constant for air (approximately 287 J/(kg·K))
T = temperature (in Kelvin)
Let's denote the velocity of sound at 0 degrees Celsius as v_0. We want to find the temperature T at which the velocity of sound is 1.25 times v_0.
So, we have the equation:
1.25 * v_0 = sqrt(gamma * R * T)
To isolate T, we need to square both sides of the equation:
(1.25 * v_0)^2 = gamma * R * T
Simplifying:
1.5625 * v_0^2 = gamma * R * T
Now, divide both sides of the equation by gamma * R:
T = (1.5625 * v_0^2) / (gamma * R)
Substituting the known values for gamma (1.4) and R (approximately 287 J/(kg·K)), we can calculate T.
Note: Remember to convert temperature from degrees Celsius to Kelvin by adding 273.15.
Let's assume the velocity of sound at 0 degrees Celsius is v_0 = 331.4 m/s (which is the approximate value).
T = (1.5625 * 331.4^2) / (1.4 * 287)
Calculating the above expression gives us the temperature T at which the velocity of sound in air is 1.25 times the velocity of sound at 0 degrees Celsius.