How was this equation modelling the motion of rear wheel of car derived? I am attempting to derive a function modelling the position of the midpoint of the rear axis of a car, given the function modelling the position of the midpoint of the front axis of a car. I found the following paper:
https://www.researchgate.net/publication/222417459_Mathematical_models_for_motion_of_the_rear_ends_of_vehicles
It gives the following diagram:

And the following equation:
$$\dot{Q}(t)=\gamma(t)(P(t)-Q(t))$$
Where $P(t)$ is the position of the midpoint of the front axis of a car, $Q(t)$ is the position of the midpoint of the rear axis of a car, and $\gamma (t)$ is the speed of the car.
However, I do not know how they derived the equation. Any help in explaining it would be appreciated.
 A: The authors do not say how they derived the formula;
perhaps they thought it was too "obvious."
So we cannot know which particular argument (out of all possible arguments) they would have made.
But here is one approach.
Given that the wheels do not slip, the two wheels on the rear axle can only roll forward or backward, which means that at any instant the instantaneous velocity of the axle at the point where where one of the wheels connects must be perpendicular to the axle (that is, in the direction the wheel rolls).
This implies that the axle either moves uniformly forward or backward
(to a position parallel to its old position)
or rotates around a point (possibly a point far to the left or right side of the vehicle).
In either case, every point on the axle, not just the points where the wheels connect, must have either zero velocity or a velocity perpendicular to the axle.
In particular, the velocity of $Q$ must be perpendicular to the axle.
The vector $P(t) - Q(t)$ is perpendicular to the axle, therefore in the same direction (or the exact opposite direction) as the velocity of $Q.$
That is, $\dot Q(t)$ must be a scalar multiple of $P(t) - Q(t).$
Let $\gamma(t)$ be the scalar.
