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The time rate of change of the position of an object

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Vapour clouds and the speed of sound

Like in the figure below: I know that as the speed of the object increases to the sonic velocity (the local velocity of sound waves), these sound waves begin to pile up in front of the object, so the … For instance the aircraft achieves the velocities $2v_s$, ..., $n v_s$, where $v_s$ is the sonic velocity . …
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