# Alternative Equation of Motion

## Main Question or Discussion Point

In classical mechanics, the acceleration $\vec{a}_A$ of a particle A relative to a reference frame S (non-rotating) fixed to a particle S, is given by the following equation:

$$\vec{a}_A = \frac{\vec{F}_A}{m_A} - \frac{\vec{F}_S}{m_S}$$
where $\vec{F}_A$ is the net force acting on particle A, $m_A$ is the mass of particle A, $\vec{F}_S$ is the net force acting on particle S, and $m_S$ is the mass of particle S.

In contradiction with Newton's first and second laws, from the above equation it follows that particle A can have non-zero acceleration even if there is no force acting on particle A, and also that particle A can have zero acceleration (state of rest or of uniform linear motion) even if there is an unbalanced force acting on particle A.

On the other hand, from the above equation it also follows that Newton's first and second laws are valid in the reference frame S only if the net force acting on particle S equals zero. Therefore, the reference frame S is an inertial reference frame only if the net force acting on particle S equals zero.

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Matterwave
Gold Member
Yes, Newton's laws hold true in inertial reference frames. This is one way to define an inertial reference frame..

Yes, Newton's laws hold true in inertial reference frames. This is one way to define an inertial reference frame..

This equation of motion can be applied in any non-rotating reference frame (inertial or non-inertial) without the need to introduce fictitious forces.

The forces acting on a reference frame determines if the reference frame is inertial or non-inertial.

Looks to me that hidden behind all that junk is the assumption that there is some universal inertial frame of reference. If you work in frame S and this frame is "fixed to a particle S" than in frame S there are no forces acting on S. Everything else after that is nonsense.

I'm not sure if you're posting this because you're confused or because you want to propose this as a new way of calculating acceleration. If it's the former you seem to have answered your own question (newton's laws are valid in inertial frames which are frames with no net acceleration) but if it's the latter I don't think this is particularly new.