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First of all, I am sorry if this is a stupid question but:

I've heard that atoms interact with each other at the speed of sound (when you for instance push a chair, the atoms collide with each other in a chain reaction at the speed of sound, making the chair move).

How can then airplanes fly faster than the speed of sound without something going crazy?

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a simple experiment, push your hand with some force through water and after the speed of your hand surpasses the wave-velocity of the water, you will see (an analogy of) the "Supersonic fracture" –  Nikos M. Jun 23 at 4:33

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The speed of sound in air determines how fast wave phenomena propagate in the air through collisions between molecules, it does not determine how fast other objects immersed in the air travel.

If an object in a fluid like air begins at rest, then the speed of that object after some time is determined by the net external force on that object. Say, for example, that an airplane's engines exert a force $F$ on the plane, then the net force on the plane will be \begin{align} F - F_\mathrm{air} \end{align} where $F_\mathrm{air}$ is the force of the air on the plane, namely air resistance. As long as this quantity is positive, namely as long is the force of the plane's engines exceeds the resistive force of air, the plane can go faster and faster provided it's made of a material that can withstand the air resistance and heat that will result.

Note, this is a super-simplified model, but it suffices to get the main point across; the speed of sound in the air does not limit the speed of other objects traveling within it, only the speed of sound waves that propagate within it.

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Thank you for this explanation! Really good. –  Tokfrans Jun 22 at 22:10
    
@Tokfrans Thanks for the kind words. –  joshphysics Jun 22 at 22:12

A sample of a gas (air, for example) is composed of molecules which are in random motion, having some mean velocity. The mean velocity of gas molecules in a sample is represented by a quantity we call temperature. The speed of sound in a gas varies with temperature, i.e. the mean velocity of its constituent molecules. The faster the constituent molecules move, the faster a sample can propagate a sound wave.

An object can move through a gas at any velocity; it simply pushes the gas molecules out of the way, accelerating them as much as it takes for them to move out of the way. In a manner of speaking, no object will move faster than the speed of sound in the medium it is traveling because it will collide with, accelerate, and effectively heat up the particles it encounters, increasing the speed of sound immediately in front of the object.

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Atoms interact between each other via various forces, like gravitation or electromagnetic forces. These forces propagate at c (speed of light in void) which is much higher than the speed of sound, hence your assumption "atoms interact with each other at the speed of sound" is wrong.

Otherwise, your description of a movement of a solid (the chair) seems correct but I see no reason why this should limit the speed of the whole chair.

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check these linke en.wikipedia.org/wiki/Supersonic, en.wikipedia.org/wiki/Sound_barrier, en.wikipedia.org/wiki/Supersonic_fracture, this is related to relative motion, in general other interactions are involved and may this be a possible reason why this can happen (or maybe not) –  Nikos M. Jun 23 at 4:37

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