A projectile is travelling in an air having pressure and temperature as 8.829 N/cm2 and – 2 0C. If the Mach angle is 40 0 then find the Mach number and velocity of projectile. Take γ=1.4 and R=287 J/kg K.

Solution :

Given :

Pressure of air, $ \begin{aligned} \qquad \quad P &=8.829 \mathrm{~N} / \mathrm{cm}^{2}\\ &=8.829 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2} \end{aligned}$

Temperature of air,

$\begin{aligned} \qquad \quad T &=-2^{\circ} \mathrm{C}\\ T &=-2+273=271^{\circ} \mathrm{K} \\ \end{aligned}$

Mach angle, $ \alpha=40^{\circ}$

k=1.4,

R=287 $ \mathrm{~J} / \mathrm{kg}^{\circ}\mathrm {~K}$

The velocity of sound, $C$ is given by, $\begin{aligned} \qquad \quad C =\sqrt{k R T} \end{aligned}$ $\begin{aligned} \qquad\qquad=\sqrt{1.4 \times 287 \times 271}=330\mathrm{~m} / \mathrm{s} \end{aligned}$ Let the velocity of projectile = V $\begin {aligned} \sin \alpha =\frac{C}{V} \Rightarrow V =\frac{C}{\sin40°}\end{aligned}$ $\begin {aligned}\qquad \qquad \qquad V =\frac{330}{0.6427}\ =513 \mathrm{~m} / \mathrm{s} \end{aligned} $