What is the relation between the velocity and temperature?

Explanation

Let us consider an ideal gas whose equation of state is
PV = nRT …………(1)
where, P is pressure, V is volume, T is temperature, n is number of mole and R is universal gas constant. For a given mass of a molecules, equation (1) can be written as
`(PV)/T`= Constant ...............(2)
For a fixed mass m, density of the gas inversely varies with volume, i.e.,
`rho propto 1/V, V=m/rho`........(3)
Substituting equation (3) in equation (2), we get
`P/rho = cT`..........(4)
where c is constant.
The speed of sound in air given in equation
`v_(A) = sqrt((B_(A)/rho)) = sqrt((gamma P)/rho) = sqrt(gamma v_(T))` cna be written as
`v= sqrt((gamma P)/rho) = sqrt(gamma cT)`............(5)
Since `v propto sqrt(T)`, the speed of sound varies directly to the square root of temperature in kelvin.
Let `v_(0)` be the speed of sound at temperature at `0^(@)` C or 273 K and v be the speed of sound at any arbitrary temperature T (in kelvin), then
`v/v_(0) = sqrt(T/273) = sqrt((273 +t)/273)`
`v=v_(0)sqrt(1+t/273) -=v_(0)(1+t/546)`
(using binomial expansion).
Since, `v_(0) = 331 ms^(-1)` at `0^(@)` C, v at any temperature in `t^(@)C` is
`v=(331 0.60t)ms^(-1)`
Thus, the speed of sound in air increases by `0.61 ms^(-1)` per degree celcius rise in temperature. Note that when the temperature is increased, the molecules will vibrate faster due to gain in thermal energy and hence, speed of sound increases.