On maximum acceleration.

Here we shall determine the maximum possible acceleration of a science-fiction spacecraft.

A few starting assumptions shall be made.

• The spacecraft is accelerated by reaction thrusters at its stern.
• The reaction thrusters can be built to produce as much thrust as desired. We shall not worry about exactly how they work, just treat them as black boxes.
• The spacecraft can be approximated as a simple prism of material with a certain compressive strength.

Since the thrusters can be as forceful as we like, the limit on spacecraft acceleration comes from its structure. If the thrusters push too hard, they will crush the spacecraft they are supposed to be pushing.

A spacecraft in deep space subject to a force giving it an acceleration a is an analogous situation to a spacecraft sitting bow-skywards on a planet with surface gravity g=a.

As such, the maximum height of a column you could build on that planet is equal to the maximum length of a spacecraft you could build to withstand that acceleration. A taller column will fail by crushing. This height can be derived as follows.

σ_c is the material compressive strength
ρ is the material density
a is acceleration
h is column height
A is column cross-sectional area

The mass of the column is given by

m = Ahρ

And the pressure at its base by

σ = ma/A = hρa

The cross-sectional area unsurprisingly cancelling out.

Obviously the pressure at the base of the column is greater than at any higher point. If this pressure is less than the material compressive strength, the column - or the spacecraft - will not fail by crushing. It may still fail by buckling, but buckling requires lateral deflection. As such, I feel it can be prevented by an actively-controlled restoring force to counter the deflection before it reaches failure, or by just not making the spacecraft too slender.

Rearranging to give acceleration as a function of the other variables,

a < σ_c/ρh

This, then, is the maximum possible acceleration of a science-fiction spacecraft.

A real spacecraft will not be a homogenous block of material. To treat it as such, we can calculate its average compressive strength and density as follows.

f is the volume fraction of the spacecraft that is structure
ρ_s is the density of the structure material
σ_cs is the compressive strength of the structure material
ρ_f is the mean density of the functional parts of the spacecraft, ie everything but its structure
The compressive strength of the functional parts is assumed to be negligible

The average compressive strength and density of the whole spacecraft are given then by

σ_c = fσ_cs
ρ = fρ_s + (1-f)ρ_f

For a worked example, consider a spacecraft with the following properties

f = 0.1, ie 10% of the craft is its structure
ρ_s = 3500 kg m-3, ie diamond
σ_cs = 12 GPa, again diamond
ρ_f = 1000 kgm-3, same as water, just feels like a good value
h = 1 km, feels like a good size for a fairly large spacecraft

Then
σ_c = 1.2 GPa
ρ = 1250
a < 960 ms-1

So the maximum acceleration is about a hundred Earth gravities.

If the shape is a pyramid or cone rather than a prism, the exact formula may change, I am uncertain on what to, but the general form will remain the same.

This limit implies various consequences. Most obviously, smaller spacecraft are capable of greater accelerations than larger ones, as most would intuitively expect. Also, to obtain maximum acceleration with a given spacecraft volume, a relatively flat spacecraft, for example a classic flying saucer, will be superior to the slender designs often seen in science fiction.

If the first assumption is violated, the entire argument can cease to hold. There  is unlikely to be much benefit from trying to place engines along the flanks of the spacecraft, material failure will still occur but in shear rather than compression. However reactionless drives that create a field acting on the entire bulk of the spacecraft, or on the space it sits in, will completely nullify the equations here. With such reactionless drives, small and large craft might accelerate equally, or large ones could even accelerate harder. Shape may not factor into acceleration performance, allowing it to be determined by other considerations.

If the second assumption is violated, the specific equations become invalid, but the generic square-cube law still indicates smaller spacecraft can probably accelerate harder than larger ones, and a flat craft still has more space on its surface for engines.

If the third assumption is violated, the entire argument again may cease to hold. One way to accomplish this is for the spacecraft to not rely solely on a physical structure for its strength, but to use dynamic support methods such as some sort of forcefield to transfer the thrust from the engines forward to the bows. Such technology may reverse the situation and make long slender spacecraft actually advantageous