Question 16.19: SONIC BOOM FROM A SUPERSONIC AIRPLANE An airplane is flying ...

IDENTIFY and SET UP:

The shock wave forms a cone trailing backward from the airplane, so the problem is really asking for how much time elapses from when the airplane flies overhead to when the shock wave reaches you at point L (Fig. 16.38). During the time t (our target variable) since the airplane traveling at speed v_S passed overhead, it has traveled a distance v_St. Equation (16.31) gives the shock cone angle α; we use trigonometry to solve for t.

\sin \alpha=\frac{v}{v_{\mathrm{S}}}                       (16.31)

EXECUTE:

From Eq. (16.31) the angle α of the shock cone is

The speed of the plane is the speed of sound multiplied by the Mach number:

From Fig. 16.38 we have

EVALUATE: You hear the boom 20.5 s after the airplane passes overhead, at which time it has traveled (560 m/s)(20.5 s) = 11.5 km since it passed overhead. We have assumed that the speed of sound is the same at all altitudes, so that α = arcsin v/v_S is constant and the shock wave forms a perfect cone. In fact, the speed of sound decreases with increasing altitude. How would this affect the value of t?