Supersonic airplane emits a sound when right over a listener

[krvolok]

Hello,

I'm having a bit of a problem with the following problem:
A supersonic airplane flying at the velocity v is flying horizontally to the ground with altitude h. At the exact moment when the plane is vertically above a listener on the ground, it emits a sound. It takes Δt seconds for the sound to reach the listener. What is the velocity of the airplane if the speed of sound is c.

The solution given is v=c/√(1-(c*Δt/h)^2). It's short so I'm not using latex :)

Anyway, my reasoning is that the sound will go straight down with speed c, since it can't go any faster. But then, this height h=c*Δt, and if you try putting that in the answer bad things happen.

Could someone explain why h =/= c*Δt and how to get that weird proportion. At least give me some hints :)

Thank you very much!

 

[SteamKing]

The sound emitted by the supersonic plane cannot travel directly to the ground. Remember, the plane is flying faster than sound can travel. If one were to look up at the moment the plane passed overhead, the plane would be visible (assuming clear skies, etc., etc.) but there would be no sound audible.

The sound from the plane cannot be heard until the Mach cone generated by the plane passes the observer on the ground.

For more info:

http://wordpress.mrreid.org/2012/12/15/sonic-booms-and-mach-cones/

http://www.grc.nasa.gov/WWW/K-12/airplane/machang.html

 

[krvolok]

But the plane is assumed to be perfectly silent, and then emits a bang. Obviously the bang will lag behind the plane since the sound can't go any faster, but how is the vertical component affected at all?

 

[krvolok]

Here, take a look at this nice applet: http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=457.0

Drag the little ear all the way left. Doesn't it travel straight down?

 

[gneill]

It appears to me that the problem is misstated. I've seen similar problems where the scenario has the sound from the plane reaching the listener at the instant the plane is overhead. Then the problem is solvable since you have a race between the plane and the sound along linear paths at fixed velocities.

As the problem stands nothing can be said about the speed of the plane. It could disappear entirely after emitting the sound overhead and no information in the problem would change. It could stand still overhead and still no information would change.