I recently saw a post (that was pretty old) asking why we don't consider the jerk function more often in sciences that deal with impacts or sudden accelerations. The general belief was that impacts, as functions of time, do not play nice and are not smooth functions on a scale where the sudden acceleration is apparent. Another observation was that sudden accelerations producing "jerk" tend to also produce oscillations whereas simple accelerations do not.
Over the course of my career I have reviewed the data from hundreds of airbag modules from vehicles involved in real-world traffic collisions. Among these data are some that are typically organized into graphs showing acceleration (typically in units of g) as a function of time over 250 ms or so in intervals of 1 ms. What you begin to notice is that impacts of a sufficient force produce clear oscillations in the acceleration data, which occur due to the materials used to construct vehicles and the type of impact (e.g., sideswipes produce less oscillations than head-ons).
This got me to thinking what would be required to actually be able to analyze the jerk of a real-world event like a collision. Another thing that is apparent about the data is that the accelerations tend to be very sharp at their maxima and minima, which makes them difficult to define as "nice" functions.
Typically, I am interested in the delta-v of a crash (among other things) so I either find a integratable function that will closely approximate the crash impulse graph and integrate, or if that is not possible, I will use numerical methods (like the trapazoid rule) to directly calculate the delta-v from the data. Usually the time increments are small enough that either way will provide useful approximations. But delta-v is far from the whole story. I think that the oscillations that a vehicle experiences are directly related to the injuries sustained by the occupants.
The question I have is, can a jerk function somehow provide insight into the oscillations that occur during a vehicle crash? If so, can a spring constant or other parameters for this oscillation be estimated using such methods? If a 3rd order non-constant function for the motion of a body can be determined, does that necessarily mean that the body was exposed to a sudden acceleration? That seems counter-intuitive.