Is speed of sound same for all inertial frames? For instance: In the diagram below suppose A shouts at B who is at a distance of S apart while the platform moves in a uniform velocity V1. If the speed of sound in air at *rest is V, wouldn't the time it takes for the sound to travel from A to B is, t = S/(V-V1).
Intuitively, I suppose an observer on earth would see sound from A has travelled further than S distance forward to reach B since B is also moving forward, so that isn't it logical to think it takes more time for sound to reach B than it should be when the platform is at rest.
I guess that, in relative to B the speed of sound is V-V1 and in relative to the earth the speed of sound is V. Am I correct?

 A: You are correct. Since A and B are at rest with respect to eachother, they will hear the same frequency. But an observer on the ground (say C) will hear a frequency which is either greater or less depending on whether she is in front of the cart or behind. In the former case, we would get
\begin{align}
& \lambda_C = \lambda_B \\
& f_C^{-1} = \frac{v_B}{v_C} f_B^{-1} \\
& f_C = \frac{v}{v - v_1} f_B.
\end{align}
This checks out because if $v_1 = v$, the sound speed and therefore the frequency observed by B is zero. This makes the finite frequency heard by the stationary observer infinitely larger.
A: The velocity of the sound relative to the observer depends on the observer's velocity, but "the speed of sound" is generally understood to mean the speed of sound relative to the medium.
A: No, the speed of sound is not the same everywhere. There is a special frame, and that is the special frame in which the air itself is at rest.
This has rather profound results when the speed of the platform V1 exceeds the speed of sound V. The term V-V1 becomes negative, because the observers are now moving at supersonic speeds.
A: You're right, the speed is not the same.  Extending your thought experiment with measurement equipment may help to understand in more detail.
Consider a sound receiver which activates a light that the sender can see.  The local propagation speed of light in air is near $c$ (and $>> V$) so we can consider the light speed signal instantaneous for the purposes of measuring the speed of sound between the two points on the platform.  This avoids a need to invoke outside observers, letting these observers directly measure the speed of sound.
(The signal could be a momentary chirp, with the receiver lighting up an LED after detecting a few oscillations at that frequency.  Or it could be a modulation (such as AM or FM) of a carrier audio frequency with the receiver demodulating, and encoding that info onto an LED in some form.  By sending a continuously varying signal, the sender can detect how far "away" in milliseconds of propagation time the receiver is.)
Note that unlike an ultrasound ranging device (e.g. with a piezo to chirp and then detect the return), the sound is only propagating one way, with the other way carried over light.
At 90% of the speed of sound in the forward direction, the sound waves aren't going much faster through the air than the platform (and sender / receiver), and sound will take about 10x longer to get there.  (Tending towards infinity in the limit as you get closer to the singularity in your formula.)
In the other direction, at close to Mach 1.0 sound will take close to half the time.  In theory you can make the propagation time arbitrarily short by going many times the speed of sound, but hearing anything over the sonic boom will be problematic.  And if you get into relativistic speeds in air, you will have bigger problems (xkcd).
Connor's answer shows that the speed of the source and receiver relative to the air can cancel out for subsonic cases, so you hear the same frequency. But that's not at all the same thing as a constant speed of sound.

Unprotected humans will have a Bad Time at transonic speeds (just below Mach 1), so let's assume sender and receiver are androids, or that the humans left a computerized experiment set up and wisely got off.  We're also neglecting that air having to go around things will have to speed up a bit, potentially going supersonic.
This is why planes not designed to break the sound barrier will also have a Bad Time as flows around parts of the aircraft become supersonic.  One of the speed limits for normal passenger jets is a maximum Mach number of typically 0.85 or so for the plane as a whole, which keeps airflow over control surfaces and wing roots comfortably below Mach 1.0.
All these speeds are of course relative to the air, which might be moving relative to the ground (aka non-zero wind speed).  Mach numbers are implicitly air speed, so yes, the air itself is a special frame of reference, relative to which true air speeds are measured.
