There are definitely situations in materials science and mechanical engineering when jerk is more important than acceleration as a factor in causing damage. A term I've seen used is "load rate." This can refer to either $dF/dt$ or $da/dt$, which differ by a factor of $m$. You'll see the acronyms ALR and ILR for average and instantaneous load rate.
A steady force can't cause wave excitations, but a varying force can. For example, when you're machining something on a mill or lathe, jerk produces "chitter," which can spoil your work. Engineers designing cams work very hard to minimize the jerk of the cam follower: "Remember also that jerk translates to an impulse and excessive impact ultimately leads to scuffed and pitted cam follower." (Blair 2005)
I know of a couple of good examples involving the human body. In crewed spaceflight, astronauts are exposed during a launch not just to high accelerations but also sometimes to what's known as a "pogo," which means an oscillating acceleration in the longitudinal direction. A pogo with an amplitude as small as $0.5g$ can apparently cause extremely unpleasant sensations in the eyeballs and testicles, as well as heating of the brain and viscera (Seedhouse 2013). Heating is a phenomenon you can't get from a static force.
Another human-body example involves running injuries. Measurements using accelerometers attached to runners' feet, legs, or hips show that during a stride cycle, there are typically two different peaks, an impact peak and another "active" peak that occurs during propulsion. The impact peak has a smaller acceleration but a larger jerk, and seems to be the factor that causes injuries: "increased impact loading was associated with an elevated risk of sustaining a running injury while peak vertical force was not." (Davis 2010)
G. P. Blair, C. D. McCartan, H. Hermann, "The Right Lift", Race Engine Technology, Vol. 3 lssue 1, August 2005
Irene Davis, quoted in http://lowerextremityreview.com/news/in-the-moment-sports-medicine/impacts-spell-injury , 2010
Erik Seedhouse, 2013, Pulling G: Human Responses to High and Low Gravity
Based on information in the comments, we are going to assume that the acceleration increases linearly with time, where the slope of this line is given by the maximum jerk $j_{max}$. Therefore:
$$a(t)=j_{max}t$$
Since $a=\dot v$, and $v=\dot x$, we can solve for $v(t)$ and $x(t)$ assuming the end-effector is moving in a straight line with $v(0)=0$ and $x(0)=0$. We will neglect the maximum acceleration and maximum speed for now. If we never reach these thresholds in the calculations, then we do not need to consider these maximum values. If we do reach these maximum values then more work will need to be done.
With the above assumptions:
$$v(t)=\int_0^ta(\tau)d\tau=\frac 12j_{max}t^2$$
$$x(t)=\int_0^tv(\tau)d\tau=\frac 16j_{max}t^3$$
Solving for $t$ when we reach the end position
$$t=\left (\frac{6x_{end}}{j_{max}}\right )^{1/3}=\left (\frac{6(28.28 \space mm)}{20\space mm/s^3}\right )^{1/3}\approx 2.04 \space s$$
Now let's check our acceleration and speed at this time:
$$a(2.04\space s)=40.8\space mm/s^2<a_{max}$$
$$v(2.04\space s)=41.6\space mm/s>v_{max}$$
So you can see that we hit maximum speed after this time interval. Therefore, we need to determine when we hit the maximum speed:
$$t_{v_{max}}=\left(\frac{2v_{max}}{j_{max}}\right )^{1/2}\approx 1.73 \space s$$
At this time I am assuming the acceleration goes to $0$? So therefore after about $1.73\space s$ you need to switch to the object moving at constant speed, which it seems like you already know how to do.
Best Answer
Acceleration does not kill us any more than speed. If your head and feet do not move at the same velocity long enough, whatever the cause, you are in trouble. Velocity does not kill us when the whole body has the same velocity.
Similarly, I doubt acceleration kills us when all parts of the body accelerate, but without having to transmit forces. It is said in a comment:
The sudden stop kills you because the deceleration (negative acceleration) that stops you is actually caused by a force transmitted through your body which cannot withstand it. The acceleration throughout the fall, no matter how strong, which applies uniformly to the whole body will not hurt it: you are in free fall.
If the same acceleration were produced by the pull of an engine attached to your feets and pulling your whole body (even without friction), rather than gravity applied uniformly to every atom of your body, your body could well be torn to pieces.
I am no expert on jerk, but I somehow doubt that it is any more danger, despite contrary statements in this accepted answer and this comment
The human body uses bones and muscles to maintain its integrity while transmitting forces. The problem of jerk is that it changes the values of forces, thus requiring muscles to adapt constantly.
But free fall satellite motion does have jerk, since the direction of gravity is constanly changing, and its magnitude depends on distance. This is generally true of non uniform gravity field.
I think, a good way of understanding what can hurt us is to model the human body as two masses, head and feet, joined with a spring. If the distance between the masses changes by more than, say, 5%, the human model is considered dead. Now, if you add a strong structure, some kind of G-suit, that forcibly preserve the distance between head and feet, thus carrying all forces that need to be transmitted, then the human model is pretty safe.
Note that submitting the head and feet to different acceleration can have undesirable effects if the difference is important. But if the body is strong enough, it can sustain small differences which it compensate with internal cohesion forces. So one might say that speed can be more dangerous than acceleration, when it is an issue of uniformity across the body.
To place these issues on the level of personal experience: we do not feel speed, but we do not feel acceleration either, or jerk. What we do experience is forces propagating through our body, when our body accelerate because it is submitted to forces applied only to some parts of it, rather than uniformly. We experience the tension of the muscles that preserve our body structure against these forces. And we perceive jerk as a need to adapt muscle tension.