You are only missing the Laws of Applied Physics.
e.g. energy conversion with the moving mass and inertia of the motor+ rotating mass to the servo power required.
This is only a "hand waving" answer not intended to be a classroom theoretical training for proper motor servo design.
Converting rotary to linear, motor+load mass inertia p=mv and motor acceleration F=ma = k*I= k * V/DCR for some motor constant k and current limited by I=V/R for some total series R including the DCR= DC winding resistance + driver resistance, Ron at DC=Vol/Iol from datasheet or some linear V/I =R drop from feedback control voltage and series current.
No load motor speed is proportional to voltage applied or generated.
i.e. kV/RPM or volts per radian/s or per RPM.
The bottom line is that the acceleration must be controlled or the heat dissipation must be removed to support the acceleration and braking of the servo loop mechanical energy into electrical energy and conduction losses in series.
The apparent motor voltage V = ( Vdc + Vbemf) where the generated back EMF voltage Vbemf=Vdc at some speed with no load or acceleration. i.e. servo motor has Vdc applied no load then open loop voltage as a generator is the same voltage. then when reverse voltage is applied you get twice the voltage apparent inside the servo motor at that instant.
Thus for any given DC servo motor when you change directions abruptly, the motor sees 2x Vdc/DCR and the power dissipation in any loop current series resistance ( such as the driver transistors) Pd=I^2 Ron with an associated temp rise * Rja [C/W] reduced by a possible heatsink.
Can you define all your variables with your understanding of Physics?. e.g. Vdc, I , DCR, driver R, mass, velocity, acceleration position error.
There must be a closed loop acceleration limit, max velocity spec and a closed loop position error curve to design the transfer functions (2). Then consider the current as acceleration feedback and voltage as a speed at a steady state then position error on arrival to servo closed loop resting with minimum overshoot and shortest possible seek time.
So there are several standard ways to design this with a PID loop but better to use linear feedback for acceleration=current and linear feedback error for the position. Then integrate only if the linear loop gain error is too high. Integration causes overshoot in the step response. Intermediate lead-lag compensation filters then optimize the servo design.
The phase-lead RC filter is partial D or derivative and I is partial low pass filter or integrate and P is the proportional loop gain which reduces the feedback error in steady state.
My rule of thumb for Servo Design, is always focus on linear feedback 1st ( the P in PID) for whatever you want to control whether it is acceleration, velocity, or position error, or for that matter temperature, vibration, etc etc. in any Control System design.
You regulate acceleration with current feedback, RPM with voltage control or tach feedback, and position with position feedback.
You can always differentiate sensors to get acceleration from velocity of a tach feedback and combine with current feedback. So choose your feedback to be linear or the P in PID loop. e.g. tach sensor for RPM velocity.
Advanced servo design uses "Vector feedback"
Then 2nd, the steady-state error and step response or settling time constraints can be optimized by tuning the "ID" in PID feedback using a variety of "control system theory methods" such as Root Locus, Nyquist methods, Body Plots etc.