My $.02
I've been sort of following this discussion, and think perhaps a fresh look is in order. I'll make two observations.
1) This problem is not about the car moving backwards because of some elastic extension of tires and brake pads. It is about the occupants moving forward during the braking phase and moving backward when the car stops. (I assume Gokul43201 correctly observed that the OP was mistyped, and the motion in question is the backward motion of the occupants relative to the car once it stops decelerating.)
2) Talk about "natural seating position" and references to equilibrium positions are, in my opinion, not helpful and not particularly relevant. The occupants of the car are not oscillators subject to some position dependent restoring force.
What has been correctly observed by some is that the occupants have to be stopped by some force (seatbelts, windshield, etc., have been mentioned) other than the direct force stopping the car, but the actual force that stops the people (in most cases) has barely been mentioned, though it was touched on here
brewnog said:
The force pulling the heads (and torsos!) back is done by the human body.
Before there were seatbelts people still managed not to crash into their windshields. They did this by applying a force to hold their bodies in a rigid configuration with pressure from the seat and floor applying a torque that keeps them from rotating and a frictional force by seat and floor, or pressure on feet against the firewall that decelerates their center of mass. Human muscles account for the torque that stops the rotation of the head and torso about the hips, and of the head about the neck.
The reason the occupants move forward in a panic stop is because they are
out of phase with the forces stopping the car. They don't get their force working until there has been a forward displacement or rotation that they manage to counter eventually to maintain a stable configuration relative to the car. And the reason they move backward is because they don't stop applying this force in phase with the forces acting on the car. They keep applying the torque to their upper bodies and the push against the seat and floor because they do not anticipate the end of the car's deceleration. There is no "equilibrium position" that they fall back into. The occupants do what they have to do to decelerate, and they keep on doing it for a while after the need is gone. They simply over-react and produce a force/torque that pushes them backward relative to the car and straightens the torso once the car stops, or greatly reduces its deceleration
Every good driver in the world learns to control deceleration to give self and occupants time to adjust their body forces to maintain their seating position without jerking around. Pay attention to yourself the next time you drive and you will notice that you gradually release pressure on the brakes as you stop to gradually lower the rate of deceleration instead of having it abruptly end. There is a reason why the rate of change of acceleration is called "jerk."