An aeroplane is to move at Mach number of 1.5 at altitude of 1000 m. The atmosphere pressure and densities at this elevation are 89.89 KPa (abs) and $1.112 kg/m^3$ respectively. Calculate the speed of the plane in Km/h at this altitude. Assume ratio of specific heats k = 1.4.

Solution:

Given :

Height, $Z= 1000 m$

Pressure, $P=89.89 KPa$

Density, $\rho = 1.112 kg/m^3$

Mach number, $M= 1.5$

Specific heat constant, $k = 1.4$

From equation of state

$$ \frac{P}{\rho}=RT$$

$$\begin{aligned}\therefore T &=\frac{P}{\rho R}\\ &=\frac{89.89×1000}{1.112×287}\\ &=281.66~K \end{aligned}$$

$$ \begin{aligned}C &=\sqrt{kRT}=\sqrt{1.4×287×281.66}\\ &= 336.4 ~m/s \end{aligned}$$

Mach number

$$ \begin {aligned}M =\frac{V}{C}\Rightarrow V &=M×C =1.5 × 336.4 \\ &=504.6 ~m/s \end{aligned}$$

Speed of plane

$$V= 1814.4 ~km/hr$$