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## Main Question or Discussion Point

I have been asked by someone if it is true that in general, for a constantly accelerating reference frame, i.e. a

My response was

I then proceeded to show this mathematically, by noting from Newton's 2nd law, that the acceleration of an object is proportional to the applied force. I iterated that Newton's 2nd law is not a definition of force, i.e. in general ##f\neq ma## (for example, Coulomb's law states that ##f=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}##, and this mathematical expression defines the force acting on a charge due to the presence of another charge), but it is the statement that the rate of change of momentum is equal to the applied force in a special set of reference frames, so-called inertial frames.

However, in this case, the apparent force acting on the particles in the non-inertial reference frame (in the absence of any actual external forces), is an artifact of the acceleration of the reference frame itself, and hence given that it can be shown, that in general for a constantly accelerating reference frame, $$\mathbf{f}'=\mathbf{f}+m\mathbf{a}_{0}$$ where ##\mathbf{a}_{0}## is the acceleration of the non-inertial reference frame, ##\mathbf{f}'=m\mathbf{a}'## are the net forces acting on the particles as observed in this frame, and ##\mathbf{f}## are the net forces acting on the particles as observed from an inertial frame.

From this we can define the fictitious force, ##\mathbf{f}_{fict}## as $$\mathbf{f}_{fict}=-m\mathbf{a}_{0}$$ hence we see that if there are no net external forces acting on the particles, then relative to the non-inertial reference frame we will have that $$-m\mathbf{a}_{0}=\mathbf{f}_{fict}=\mathbf{f}'=m\mathbf{a}'\Rightarrow \mathbf{a}'=-\mathbf{a}_{0}$$ and so, relative to an observer in the non-inertial reference frame, all particles will appear to accelerate in the opposite direction to the acceleration of the frame, independently of their mass.

I think I've correctly informed them, but I'm now doubting myself. I don't want to convey incorrect information and would appreciate someone taking a look at this and letting me know what they think (importantly, letting me know if anything is incorrect about it).

*non-inertial*reference frame, the acceleration of a particle (as observed in this frame) due to the corresponding fictitious force is independent of its mass.My response was

*yes*. My reasoning being that the apparent acceleration of any object within the accelerating frame of reference is due to the fact that the frame itself is accelerating, and not the objects themselves (neglecting any other external forces that may be acting on them). As such, relative to this*non-inertial*reference frame, all objects within the frame will appear to accelerate in the opposite direction*at the same rate*. For example, if a car is accelerating to the rate at a constant rate, an observer in the car would observe objects at the front of the car accelerate towards the back of the car at exactly the same rate, independently of their respective mass, such that they will hit the back of the car at exactly the same time (ignoring any air resistance).I then proceeded to show this mathematically, by noting from Newton's 2nd law, that the acceleration of an object is proportional to the applied force. I iterated that Newton's 2nd law is not a definition of force, i.e. in general ##f\neq ma## (for example, Coulomb's law states that ##f=\frac{1}{4\pi\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}##, and this mathematical expression defines the force acting on a charge due to the presence of another charge), but it is the statement that the rate of change of momentum is equal to the applied force in a special set of reference frames, so-called inertial frames.

However, in this case, the apparent force acting on the particles in the non-inertial reference frame (in the absence of any actual external forces), is an artifact of the acceleration of the reference frame itself, and hence given that it can be shown, that in general for a constantly accelerating reference frame, $$\mathbf{f}'=\mathbf{f}+m\mathbf{a}_{0}$$ where ##\mathbf{a}_{0}## is the acceleration of the non-inertial reference frame, ##\mathbf{f}'=m\mathbf{a}'## are the net forces acting on the particles as observed in this frame, and ##\mathbf{f}## are the net forces acting on the particles as observed from an inertial frame.

From this we can define the fictitious force, ##\mathbf{f}_{fict}## as $$\mathbf{f}_{fict}=-m\mathbf{a}_{0}$$ hence we see that if there are no net external forces acting on the particles, then relative to the non-inertial reference frame we will have that $$-m\mathbf{a}_{0}=\mathbf{f}_{fict}=\mathbf{f}'=m\mathbf{a}'\Rightarrow \mathbf{a}'=-\mathbf{a}_{0}$$ and so, relative to an observer in the non-inertial reference frame, all particles will appear to accelerate in the opposite direction to the acceleration of the frame, independently of their mass.

I think I've correctly informed them, but I'm now doubting myself. I don't want to convey incorrect information and would appreciate someone taking a look at this and letting me know what they think (importantly, letting me know if anything is incorrect about it).