Classical Mechanics/Non-Inertial Reference Frames
It is very important to acknowledge how to construct equations inside of an inertial frame of reference. (As even the Earth is a non-inertial frame)
Consider an inertial reference frame S and a second reference frame S0 which is moving with respect to S with a velocity and accelerating with respect to S at a rate .
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 where 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 . Using this fact we can differentiate ( ) and then substitute the force in the inertial frame ( ) to get an expression for the force measured by an observer in the non-inertial frame : .
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 :