Two frame of reference A and A'. A' starts accelerating with respect to A. The distance of separation of the two frame of reference is s = 1/2 at^2 x' = x - s = x - 1/2 at^2 Differentiating twice with respect to time we get d^2x'/dt^2 = d^2x/dt^2 - a d^2'x/dt^2 + a = d^2x/dt^2 Therefore F' = m(a + d^2'x/dt^2) and F = ma. Is Newton's Third law symmetrical in a Galilean accelearating reference frame? The two formulas are different, but since acceleration is a vector quantity, which means is simply the resultant acceleration for A'. So is Newton's third law symmetrical in an accelerating Galilean frame of reference? And how can we measure the acceleration if we are inside the accelerating frame of reference. It is a common experience that people tend to be pushed back to the seat when the car is accelerating, is it possible to observe the change of motion of a body in an accelerating frame of reference while you are being pushed back at the same time?