Bidirectional jerk motion on a stopping vehicle A stopping vehicle (say a car) has an apparent retardation (which may/may not be constant in magnitude) when force via brakes is applied. 
I travel by subway trains, and I noticed an odd phenomenon. The thing about such trains (might be irrelevant) is that, being light-weight, their motion somewhat mimics that of cars, and the effects of motion are more apparent as one usually stands in such trains. The thing I noticed was that I could feel a  force pulling me in the initial direction of motion, as the train slowed down. That obviously is the inertia. But as soon as the train halted, I noticed a secondary jerk...this time in the opposite direction, and it was kind of short lasting. 
I'm curious to know, what causes this secondary jerk backwards as soon as a vehicle comes to rest. I'm guessing it has something  to do with the reaction force by the brakes which overcome the forward motion and provide an impulse backwards. But then it has to have a proper force 'mirror' as per the third law of motion. Also, there never is any intentional backwards motion here (the drivers are precise, I guess). 
So what could it really be?  
 A: I have noticed this effect often in cars and sometimes in trains.  This is the reason I think it happens though I can't claim to have done any research.
The car stops because the breaks are applied, the wheels stop turning and there is a force of static friction between the road and the tires.  In the car frame of reference I experience this backwards acceleration of the car as a forwards (inertial) force on me.  
The reason I then feel a secondary force backwards must be because the car has a short acceleration forwards near the end of its stopping.  I assume this is because, as it stops, there is some energy stored as elastic potential energy in the tires and/or the suspension system.  Once the car is finished moving forwards this stored elastic energy causes it to "spring back" a bit right at the end.
Subway trains don't usually have rubber tires (though they do in Montreal) but the do have some sort of sprung suspensions system that could act in a similar way.
A: "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.
