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Well... It is certainly possible in some cases to show that you are not in
a Newtonian inertial frame, but it is *not* possible to show that you are
even approximately *in* one.
It can be an "appropriate approximation" to
consider the earth's surface a Newtonian inertial frame even if it is
*nothing at all* like one.
Consider this: To establish that a reference frame is a Newtonian inertial
frame, you need to demonstrate that it is not accelerating with respect to
other Newtonian inertial frames, e.g., those that are free floating in
deep space.