# Accelerating at $\rm 1,000,000 \, m/s^2$ to $\rm 1 \, m/s$?

What would it look like if someone was accelerated at $\rm 1,000,000 \, m/s^2$ to $\rm 1 \, m/s$? Would they die? Would their body stay intact? My guess is that the answer might be more interesting if we make the assumption that not all parts of the body are accelerated uniformly as you might find in a freefall situation?

It's entirely possible that the answer to this question is highly trivial, but when I asked my physics-oriented friends, they all seemed to disagree with each other. Some said that it would be similar in effect to someone dying after smacking into the ground after a long fall, while others claimed that in fact the situation was different from this in subtle ways. Thoughts on any of this would be greatly appreciated!

• Their body will remain intact, as there's no relative difference in acceleration between any two points on the body, and hence there are no stresses applied on the body. A tremendous amount of power is however, needed by the force to have such high acceleration. – Pritt Balagopal Nov 9 '17 at 5:30
• Related physics.stackexchange.com/q/200268/26076 and the dupe question. See my answer to the first. I discuss the uniformly applied arbitrarily high acceleration - basically the same conclusion as Asher's answer below - in the last paragraph – WetSavannaAnimal Nov 9 '17 at 9:53
• @PrittBalagopal If the guy was pushed from the back his front would have a different acceleration then his back. The OP is more interested in "if we make the assumption that not all parts of the body are accelerated uniformly as you might find in a freefall situation". – J. Manuel Nov 9 '17 at 14:01

Probably you wouldn’t even notice

The thing is that if all points of a body are equally accelerated then all points of that body will retain their relative positions and therefore it will keep its shape and integrity. The problem starts when you have differential accelerations, i.e., different parts of the body are moving differently. For example:

Let’s consider a guy with his back in with cross contact with a wall. Suddenly, the wall moves, pushing him forward with an arbitrary acceleration and time. Since the wall is in contact with his back, this will start moving first and only after a certain time the chest will follow. This will cause different displacements on his back and chest. Let’s imagine that after the acceleration his back have been displaced by 40 cm and his chest by 30 cm. The result will be the same as been crushed by a car crusher for 10 cm. The reverse of this happens to someone falling from a height. The part that makes the first contact with the ground stops but the opposite side continues moving crushing him. Therefore, just as in a car crusher, the key here is displacement

The displacement caused by a constant acceleration (force) is $$\rm d= \frac{1}{2}at^2 \tag{1}$$ and the time elapsed during acceleration is $$\rm t=v/a \tag{2}$$ meaning that, knowing the final speed $\rm v$, the displacement caused by a constant force is $$\rm d=\frac{v^2}{2a} \tag{3}$$ which in your situation $$\rm d=\frac{1^2 \,m^2/s^2}{2•10^6 \,m/s^2}=0.5•10^{-6}m=0.5 \,µm$$ In other words, you will end up with a half micrometer amplitude chock wave bouncing in your body and dissipating.

A cool result from equation (3) is that contrary to the expected result, if you keep the same end speed $\rm v$, the higher the acceleration, the harmless it gets, to the point that an infinite acceleration would cause no harm at all. This happens because displacement is quadratic with time and only linear with the acceleration. To keep the same end velocity, with higher and higher accelerations, one would need a quadratic smaller time, resulting in smaller displacements.

• You made a good point that I forgot to mention. Higher acceleration means less damage if any (giving that final speed). It is quite counter intuitive. .. – Alchimista Nov 9 '17 at 19:15
• This is the answer I was looking for--fascinating result! – user8378087 Nov 11 '17 at 1:02

If all parts of the body are accelerated uniformly, for example by a uniform (non-tidal) gravitational field, there is no direct effect on the body. Acceleration itself has no direct effect on physical structures; accelerometers, including the natural ones built into the human body, rely on non-uniform acceleration to work.

In fact, the only reason high accelerations such as g-forces and impacts are harmful to physical structures (including the human body) is because differential accelerations cause stresses in the material which can damage it. Hard shocks caused by high acceleration can cause plastic deformation in otherwise elastic materials, such as people. It's not the total velocity that matters, it's the magnitude.

This would a body at more than 100000 g. I think the answer might seem obvious. All tissues within the body will move at different speeds according to their density. The body won't splash like against a wall but it will be internally devastated at a small length scale.

However, the body displacement is less than a micrometer so that it might be possible that the body can resist and accommodate the effective internal displacement of its differently rigid and dense organs and tissues.

I am afraid that we cannot answer without a knowledge of anatomy/physiology.

If this answer/not answer do not conform with the SE rules I will delete it and put in comment(s).

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