Classical Mechanics/Non-Inertial Reference Frames

Consider an inertial reference frame S and a second reference frame S0 which is moving with respect to S with a velocity ${\displaystyle {\vec {V}}}$ and accelerating with respect to S at a rate ${\displaystyle {\vec {A}}}$.
From the inertial reference frame (S) Newton's second law will hold and any object of mass m will be observed to have a force acting on it of ${\displaystyle {\vec {F}}=m{\ddot {\vec {r}}}}$ where ${\displaystyle {\vec {r}}}$ is measured from the origin of the frame S.
From the non-inertial frame (S0) we must relate the quantities using the Galilean transformation for a moving reference frame, so that the velocity of the mass in the new reference frame is ${\displaystyle {\dot {\vec {r_{0}}}}={\dot {\vec {r}}}-{\vec {V}}}$. Using this fact we can differentiate ( ${\displaystyle {\ddot {\vec {r_{0}}}}={\ddot {\vec {r}}}-{\vec {A}}}$ ) and then substitute the force in the inertial frame ( ${\displaystyle {\vec {F}}=m{\ddot {\vec {r}}}}$ ) to get an expression for the force measured by an observer in the non-inertial frame : ${\displaystyle m{\ddot {\vec {r_{0}}}}={\vec {F}}-m{\vec {A}}}$.
The conclusion that we can reach is that we may continue to use Newton's laws in the non-inertial frame, so long as we add the additional "force" due to the motion of the frame, which is often called the inertial force : ${\displaystyle {\vec {F}}_{inertial}=-m{\vec {A}}}$