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HN Puzzler

5 pointsDanielBMarkham15 comments

Here's a puzzle for anybody that would like to take it up today:

I just heard a jet fly overhead. Curious, I went outside. As we all know, sound travels much, much more slowly than light, so the jet was gone from sight by the time I started searching for it, even though it still sounded like it was overhead.

Now, given the speed of sound, and a jet flying at 500 mph, at what elevation would the jet need to fly in order to create the greatest angle between where the sound from the jet is coming from and where the jet actually is?

If anybody wants to kick it around, I'll warn you up front that there are a couple of tricks to finding the solution.

EDIT: As commenter pointed out, the question is a bit incomplete. I'll change that to mean "at what elevation and distance from the observer would the jet need to fly in order to create the greatest angle between where the sound from the jet is coming from and where the jet actually is?"

Assuming we are on planet earth and standing standing still in a flat desert.

Under a number of assumptions, the plane should now be heading away from you, at an angle of about 19.2 degrees from the vertical, and the altitude does not matter. A more precise estimate of the angle is 90 * atan2(sqrt(76 * 76 - 25 * 25), 76) / atan2(1, 0).

Key assumptions are that there is no wind, the speed of the plane is a constant 500 mph, the speed of sound is constant at 760 mph (actually untrue in real life, see http://www.engineeringtoolbox.com/elevation-speed-sound-air-...), the airplane is maintaining a constant elevation, and the curvature of the Earth is irrelevant.

In that situation the actual elevation of the plane doesn't matter. Look at the triangle formed from where the airplane let out the sound, where the observer is, and where the airplane now is. If you double the height of the airplane then that becomes a similar triangle with all lengths doubled. So we may as well assume that the sound traveled for 6 minutes, so the plane has moved 50 miles and the sound moved 76.

It is easy to verify that at any point in time the rate at which the airplane is adding to the angle between where it was and where it will be depends on how close it is to you (the closer the better). From this it is easy to show that the maximum has to happen with the original position of the plane, the observer, and the current position forming an isosceles triangle. And that happens when the observer, the half-way point for the plane, and the current position form a right-angled triangle with opposite 25 miles and hypotenuse 76 miles. So the adjacent is sqrt(76^2 - 25^2).

The rest falls out of the definition of atan2. (Assuming no careless errors.)

I don't think you can assume a constant rate for the speed of sound.

Sorry, not trying to add new parameters, but the question is based on a real-world scenario. So air density has to play a factor (although it's certainly fair to assume density changes evenly with altitude instead of the much more complicated real-world situation)

I would have brought up density, but it was one of the gotchas, and I was curious to see who spotted it first.

When you allow variable air density, and variable speed of sound, the problem depends strongly on what assumptions you have made there. For example one complication is that if the plane is high in the sky at a 45 degree angle, even if it isn't moving it will sound like it is in a different place than it really is thanks to refraction.

Those variants of the problem are much harder, and less interesting to me.

Wow. Didn't think of refraction. Wonder how much the delta in appearance and sound source for a high altitude jet is due to refraction?

Well, given that with non-variable air density altitude doesn't matter, it should seem that if air density [actually, temperature] did matter (that is, it caused the sound to travel slower), then the higher the altitude, the higher the "plane to speed of sound" ratio would be, and the higher the angle between hearing the plane and observing it. So, I'd answer with infinite altitude. Eventually, the plane would travel from horizon to horizon without the sound ever reaching you.

Or, you're going to say that curvature is now a parameter, or the air temperature is some weird function of altitude, etc, etc.

No, I'm not just trying to over-complicate things for the sake of it.

My first though was eye-level, then, after thinking about air density, it was infinity.

But the problem with distance is that *as things get farther away, they appear to move slower". So although some pulsar in the crab nebula might be moving millions of miles an hour, and there is no sound, there is also no movement, therefore the angle of the sound source to the light source is nothing. No sound.

So the only other (natural) constraint I would mention would be that the sound has to be able to reach the observer.

There is insufficient data to answer this question.

Perhaps the jet is flying on the other side of cloud cover, on the other side of a mountain range, or inside a canyon. What elevation is the observer? What is the velocity of the observer to the light reflected from the jet and what is the velocity of the observer to the sound coming from the jet? What is the temperature of the air or medium that the light and sound are travelling in. How do we know it is the jet making noise?

This kind of poorly thought out question that makes me want to quit software development, which is full of users with requests like this.

Oh come off it. There is no cloud cover, the earth is a flat plane, the flight path is level, and we're assuming standard temperature and pressure. Lightspeed may be neglected, and the observer is at rest with regards to the earth and atmosphere. The jet is making noise.

Pedantic people who can't do back-of-the-envelope calculations due to "underspecified" problems piss me off, partly because I used to be one of them. ;-)

Yes. Make reasonable assumptions and document them.

You didn't see the original question. I did my best to answer it after he modified it, and I still misunderstood. Which proves my point.

How about:

Standing roughly between the apparent location of the noise and current apparent location of the light reflected from the jet with the jet flying as close to your elevation as possible, and assuming the jet is flying in a straight line over the observer on a flat plane (i.e. not the earth, which is round) and assuming we can rule out all other factors.

If the jet is flying in a straight line over the observer then the angle to the jet and sound both would be the same, ie, the jet is approaching directly at the observer (or leaving), so there could be no delta between the sound angle and the light angle.

??? I misunderstood you to mean the angle between the apparent source of the sound and the apparent source of the reflected light from the jet. Perhaps you could draw it out to explain what you are looking for.

Yes, that's what I meant.

If the jet is flying at your altitude directly at you, it appears as a speck and the speck just gets bigger as it comes closer and closer to you. The sound comes from the speck. There can be no delta.

I guess at the exact instant the jet flies over your head there would be some delta, but at that moment the jet is almost directly on top of you. Sound traveling a few inches from the jet engine to your ears, even as slow as sound is, would happen pretty quickly in comparison to the jet moving (which is slower than sound)

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