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## Main Question or Discussion Point

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

[tex]\vec{a}_A = \frac{\vec{F}_A}{m_A} - \frac{\vec{F}_S}{m_S}[/tex]

where [itex]\vec{F}_A[/itex] is the net force acting on particle A, [itex]m_A[/itex] is the mass of particle A, [itex]\vec{F}_S[/itex] is the net force acting on particle S, and [itex]m_S[/itex] 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.

[tex]\vec{a}_A = \frac{\vec{F}_A}{m_A} - \frac{\vec{F}_S}{m_S}[/tex]

where [itex]\vec{F}_A[/itex] is the net force acting on particle A, [itex]m_A[/itex] is the mass of particle A, [itex]\vec{F}_S[/itex] is the net force acting on particle S, and [itex]m_S[/itex] 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.