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One can use this formula to explore whether or not any possible measurement would remain the same in different reference frames. For instance, if the passenger on the train threw a ball forward, he would measure one velocity for the ball, and the observer on the platform another. After applying the formula above, though, both would agree that the velocity of the ball is the same once corrected for a different choice of what speed is considered zero. This means that motion is "[[invariant (physics)|invariant]]". Laws of [[classical mechanics]], like [[Newton's laws of motion|Newton's second law of motion]], all obey this principle because they have the same form after applying the transformation. As Newton's law involves the derivative of velocity, any constant velocity added in a Galilean transformation to a different reference frame contributes nothing (the derivative of a constant is zero).
This means that the Galilean transformation and the addition of velocities only apply to frames that are moving at a constant (relative) velocity. Since objects tend to retain their current velocity due to a property we call [[inertia]], frames that refer to objects with constant speed are known as [[inertial reference frames]]. The Galilean transformation, then, does not apply to [[acceleration]]s, only velocities, and classical mechanics is ''not'' invariant under acceleration. This mirrors the real world, where acceleration is easily distinguishable from smooth motion in any number of ways. For example, if an observer on a train saw a ball roll backward off a table, he would be able to infer that the train was accelerating forward, since the ball remains at rest [[Newton's first law|unless acted upon by an external force]]. Therefore, the only explanation is that the train has moved underneath the ball, resulting in an apparent motion of the ball. Addition of a time-varying velocity, corresponding to an accelerated reference frame, changed the formula (see [[pseudo-force]]).
Both the Aristotelian and Galilean views of motion contain an important assumption. Motion is defined as the change of position over time, but both of these quantities, position and time, are not defined within the system. It is assumed, explicitly in the Greek worldview, that space and time lie outside physical existence and are absolute even if the objects within them are measured relative to each other. The Galilean transformations can only be applied because both observers are assumed to be able to measure the same time and space, regardless of their frames' relative motions. So in spite of there being no absolute motion, it is assumed there is some, perhaps unknowable, absolute space and time.
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