High speed does not kill. Does acceleration do it ? or jerk? In a recent question the OP asked why high speed will not kill us.
The accepted answer, highly upvoted, stated very first that 

Speed doesn't kill us, but acceleration does.

The second answer (also well upvoted) concurs:

The danger comes from acceleration, not velocity.

Is that true?
Two comments by hdhondt and Adam D. Ruppe somewhat debunk it. but they
are only comments. They could not be answers, as the question
"Why doesn't the speed of Earth (moving through space) kill us?"  was not
about acceleration.
I wrote this question, and my answer to it (below), before I had read these
comments. Like many users, I do skip a lot of comments. When I finally
saw these comments already made, I wondered whether I should persist
with this question. The point seems important enough to justify it, considering the high popularity of this topic.
 A: babou, this is really an extended comment to your own answer.
I think your answer is pretty much spot on, but I would simply the reasoning a bit. What kills you is when the distance between different parts of your body changes. You give the example of the separation between your head and feet changing by more than 5% (something exploited by hangmen over the centuries :-).
To get the position of any particular volume element $i$ within your body you simply integrate the equation:
$$ \frac{d^2\vec{x_i}}{dt^2} = \frac{1}{m_i}\vec{F}_i(t) $$
where $\vec{F}_i(t)$ is the net force on that volume element. In a fall, collision or whatever the force $\vec{F}(t)$ varies with position in the body and that's what causes the distance to change and the subsequent injury. You can Taylor expand the force as a function of time and separate it into acceleration, jerk, and higher order terms then argue which is the more important, but I think this is a bit of a red herring. If $\vec{F}(t)$ is the same throughout the body then there will be no distance change and no injury so it's not the magnitude of the acceleration, jerk, or whatever, but their inhomogeneity that kills you.
A: It really is the stress that kills you. Velocity, acceleration & jerk are all fine as long as they are spatially uniform. It is a postulate of general relativity that you can not even detect acceleration due to a uniform gravitational field, no matter how intense. However if the there are spatially non uniform forces applied to your body, then there will be stresses: tension, compression, shear or torsion. If they are high enough, something will give. This is what happens when you get hit, cut, stabbed, shot or even hanged, drawn and quartered.
A: The problem is, there isn't just one way in Newtonian mechanics to kill someone. You can cause as little or as much acceleration as you want. A few things worth analyzing are:


*

*Whiplash. If you're under constant acceleration and you reach a steady state (and aren't dead yet), a change in acceleration (jerk) could cause a whip effect.

*The Earth-Sun system. in a sun-centered inertial frame, people on the Earth don't die despite an acceleration on them.

*A centrifuge/jet with constant acceleration. By changing reference frames and using an equivalence principle, you can look at it like the weight of your body presses down on you until something squishes/breaks, at which point you cease to be.

*Knives, where high [relative] velocity can kill. So can low relative velocity!


So, these examples I think prove that a statement like "[X] doesn't kill, it's [Y]" where [x,y] are one of acceleration/jerk/velocity, is just too general to be correct. You have to look at the whole dynamics of the situation.
A: If you fly head first into a wall, at the moment of impact the top of your head will accelerate very rapidly while the est of your body continues to travel at a constant speed until they make contact with the wall. Clearly having different parts of your body rapidly accelerate in different direction will lead to some very large forces on various parts of your anatomy and is not going to end well. This is not the same thing as jerk which would be about the difference in acceleration of a single part of your body at different times. If your entire body receives exactly the same acceleration, this will not lead to any internal strains and won't cause you any problems. In practice however the only external force capable of delivering a truly uniform acceleration is gravity, as it acts on each part of you body directly and in proportion to its mass. This means that often it is enough to simply look at the magnitude of the forces acting on your body and not look to carefully at how they are acting and treat gravity is an exception, so the the other answers are not so much wrong as being approximate. 
A: 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:

It's not the fall that kills you; it's the sudden stop at the end.

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.
