How to represent the lag/settling time of a DC motor?
Torque is proportional to current speed is proporitonal to the back-emf.

The back emf rises (as rthe speed rises) until the current allowed by the difference between back-emf and and the supply voltage exactly matches the drag caused by solid friction and aerodynamic drag.

Once you have the expression for drag at any speed, the current-torque relation, and the inertia of the motor you can predict how the motor will behave given any input voltage.