Question

the relationship between the cones vertex angle, θ, and the mach number, m, of an aircraft that is flying faster than the speed of sound is given by the formula below. if θ = 120°, determine the mach speed, m, of the aircraft.

\\( \frac { \sin \frac { \theta } { 2 } } { 1 } = \frac { 2 } { m } \\)

m = \\( \square \\)

(do not round until the final answer. then round to one decimal place as needed.)

Explanation:

Step1: Substitute the value of $\theta$

Given the formula $\frac{1}{m}=\frac{\sin\frac{\theta}{2}}{1}$, and $\theta = 120^{\circ}$. First, find $\frac{\theta}{2}=\frac{120^{\circ}}{2}=60^{\circ}$.

Step2: Calculate $\sin\frac{\theta}{2}$

We know that $\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866$.

Step3: Solve for $m$

From $\frac{1}{m}=\sin\frac{\theta}{2}$, we can rewrite it as $m = \frac{1}{\sin\frac{\theta}{2}}$. Substituting $\sin\frac{\theta}{2}=\sin60^{\circ}$, we get $m=\frac{1}{\sin60^{\circ}}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, then $m=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\approx1.2$ (after rational - izing the denominator and rounding to one decimal place).

Answer:

$1.2$