As you\'re hiking toward the Travertine Palms in the northeast corner of the Anz

Question

As you're hiking toward the Travertine Palms in the northeast corner of the Anza-Borrego Desert State Park, you notice an F-15 Eagle flying directly overhead at constant altitude, h. It seems to make no sound, so you wait for the "sonic boom" shock wave to reach you. The shock front reaches you a time t after the jet crosses your zenith. If the angle between the shock wave and the level ground is Phi, then you can show that sin Phi = v/ve, where v is the speed of sound and ve is the speed of the Eagle. (You may take this equation as given.) Show that the speed of the F-15 is given by ve = hv/

Explanation / Answer

Let sin(@) = v/ve Draw a right triangle, and apply the Pythagorum theorem opposite: h adjacent side: vet hypotenuse: L= (h^2 + (vet)^2 )^1/2 Sin(@) = h/ ((h^2 + (vet)^2 )^1/2 v/ve = h/ ((h^2 + (vet)^2 )^1/2 v^2/ve^2 = h^2/(h^2 + vet^2) v^2h^2 + v^2ve^2t^2 = ve^h^2 ve^2 (h^2 - v^2t^2 ) = v^2 h^2 ve = vh/ (h^2 - v^2t^2 ) ^1/2 Sorry about the mess.

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