Chapter 25, Problem 69A

The half-angle of the shock-wave cone generated by a supersonic aircraft is $45°$. What is the speed of the plane relative to the speed of sound?

To determine

The speed of the plane relative to the speed of sound.

Answer to Problem 69A

The speed of the plane is $1.41$ times the speed of sound.

Explanation of Solution

Given:

The half-angle of the shock wave cone is $45°$ 

Formula used:

According to hypotenuse theorem,

  $\text{Perpendicular = sin}\theta \times \text{ Hypotenuse}$

Calculation:

Consider the figure shown below.

  

AB shows the direction of the plane and AC shows the direction of the speed of sound. Thus, using hypotenuse theorem,

  $\begin{array}{l}\text{AC = sin 45}°\times \text{ AB}\\ \text{AB = }\sqrt{2}\text{AC = 1}\text{.41}\times \text{AC}\\ \overrightarrow{v_P}=1.41\overrightarrow{v_S}\end{array}$

Conclusion:

Thus, the speed of the plane is $1.41$ times the speed of sound