When a rocket is traveling toward a mountain at 100 m/s, the sound waves from this rocket's engine approach the mountain at speed V. If the rocket doubles its speed to 200 m/s, the sound waves from the engine will now approach the mountain at speed of? The answer was V. But I would like an explanation to why the speed is unaffected because I thought it would be 2V. Thank You
The sound wave is a property of the air, which does not care whether the object producing the sound was moving or not. Therefore in both cases the sound travels at the same velocity.
That is the answer your book wanted, and while it is almost precisely correct, there is a slight complication. The point is that if an object in motion emits a sound which locally as some frequency, the frequency of a sound heard by a stationary observer is different. (Think of an approaching then receding freight train; the whistle sounds high pitch as it approaches and changes to lower frequency as it recedes.)
Well, it turns out that sound in air has a dispersion relation, which is a fancy way to say that the sound velocity depends (very mildly) on the pitch. So the speed of any given frequency component the sound wave emitted by a rocket at 100 m/s will be very slightly different from the sound of that same frequency component emitted by a 200 m/s rocket.