Relative kinematics and laws of Newton

Question

I am an engineering student and currently taking a class on kinematics and dynamics. I study at a German university so it may be that I don't translate everything correctly.

In the first module of the course we have studied the kinematics of rigid bodies and relative motion. We learned how to describe the motion of a point or particle from an absolute reference frame via a moving reference frame. We would decompose the absolute position vector to a point into a vector to another reference frame(which may be moving, accelerating or rotating) plus a relative vector from the second reference frame to the point(particle). We would then do differentiation and different cross products to find the velocity and acceleration of the absolute vector via the other two. Now in the second module we are learning about newtons laws and inertial reference frames; and the big statement here being that every reference frame that is moving with constant velocity and no rotation relative to an absolute reference frame is also an inertial frame. My confusion arises here: we are using accelerating and rotating reference frames to find the velocity and acceleration of a particle in the absolute reference frame, but, then we use the acceleration that we find to find the forces on the object. This doesn't make sense to me, and seems to contradict the condition of inertial reference frames, since the reference frames we use to find the acceleration of the particles may be rotating or accelerating.

I think I have a big misunderstanding relating kinematics with newtons laws.

Answer 1

There is little 'extra' physics in kinematics beyond what can be derived using everyday calculus and geometry. In particular, all kinematical relations always hold. For example, the relative velocity between two moving objects is always the difference of the two velocities at any instant of time, no matter how the velocities came about, or how they change later.

Now Newton's laws only hold in an inertial frame. To elaborate, let us assume that there are certain known forces on a particle such as gravity, friction etc. which everyone agrees upon. Then, the resulting acceleration is proportional to the net force only in inertial frames. In the rotating, accelerating frames, you'll find that there are other effects ('fictitious forces'). The acceleration will not then be proportional to the sum of gravity, friction etc. but also of these other effects.

In summary, you can only use the second (or any other) law of Newton in an inertial frame, but because the relations of kinematics are universal, you can use kinematics to transform to/from a non-inertial frame to get the correct motion.

A remote analogy is perhaps that the common log tables are only made for base 10, but I can obtain the logarithm for any other base from the value I read off from the tables in base 10; the relations between logarithms in different bases remain true independent of whether the common log tables are applicable to those bases.