"Jerk" is indeed the correct term to describe both the experience and the cause, which is a (sudden) change in acceleration.
If the car decelerates at a constant rate, there is a constant force on you from the safety belt, to prevent you hitting the windscreen. If the car were to maintain the same deceleration when it had reached zero velocity then it would immediately start going backwards. However, the car does not move backwards. The deceleration changes from a constant value to zero in a very short time. But there is still a force on you from the springiness in the safety belt, which is under tension, throwing you back into the seat, which is no longer accelerating backwards. The force on you from the safety belt changes suddenly as you are flung back into the seat, as it did if the braking also started suddenly; the sudden change in the force on you is what causes the discomfort.
The same effect happens (in reverse) when a car accelerates. During constant acceleration your seat pushes you forward with a constant force. The seat is padded for comfort, so that, as with the safety belt, the force on you does not change suddenly. When the driver pulls his foot off the accelerator to change gear, the car and seat suddenly stop accelerating. The springy seat is still pushing you forward, so you accelerate forward away from the seat. The force on you drops to zero suddenly as you "jerk" forward. If the seat had not been padded, the change in the force on you would have been even more sudden, even more uncomfortable.
The Wikipedia article explains it this way :
A highly reproducible experiment to demonstrate jerk is as follows. Brake a car starting at a modest speed in two different ways:
- apply a constant, modest force on the pedal till the car comes to a halt, only then release the pedal;
- apply the same, constant, modest force on the pedal, but just before the halt, reduce the force on the pedal, optimally releasing the pedal fully, exactly when the car stops.
The reason for the by-far-bigger jerk in 1 is a discontinuity of the acceleration, which is initially at a constant value, due to the constant force on the pedal, and drops to zero immediately when the wheels stop rotating.
Note that there would be no jerk if the car started to move backwards with the same acceleration. Every experienced driver knows how to start and how to stop braking with low jerk. See also below in the motion profile, segment 7: Deceleration ramp-down.
The situation on the braking train is explained by muscular control (see Wikipedia article : Physiological Effects).
In place of the safety belt your muscles supply a braking force when you hold onto a rail or hanging strap to avoid falling forward. When the train reaches zero velocity but does not move backwards this force is no longer required. However, the time over which the change in force is required is too short for your muscular control system to respond to, with the result that you involuntarily throw yourself backwards.
Whereas a constant force is tolerable if not too large, a sudden change in force can be very uncomfortable. Such discomfort is exploited in roller-coasters to enhance the thrill of the ride. Not only "jerk" (rate of change of acceleration) but also higher derivatives such as "jounce" (rate of change of jerk) are desirable and carefully designed.