As gauss-building comes forward, new and new models appear with relatively high initial velocities, I think it’s time to rise the question about accuracy of coilguns. To answer this question one should at first mention the important fact: in contrast to pneumatic and firearms, the projectile in coilguns is moving inside barrel with some gap (i.e. the diameter of the projectile is a bit smaller than one of the barrel). The reason is simple – it is just a way to reduce friction which degrades efficiency of  acceleration (not so large itself).

Let us consider the gap as δ, and its relation to the diameter of projectile d as ∆, and look how these values influence the accuracy of shooting. At first let’s analyze a cylindrical projectile with length of l:

Fig. 1.

The picture shows that the projectile leaving the barrel with deviation angle α has  a gap between his rear end and the barrel, which is calculated as:

f = l ·sin α = D - d ·cos α                                            (1)

Then, considering small α (i.е. cos α ≈1), we easily get

α = arcsin (δ/l)= arcsin {∆(d/l)}                          (2)

The same situation for spherical projectile (fig .2):

Fig. 2.

Now the trajectory of maximal deviation will be the tangent to the sphere crossing the end of barrel, and deflection angle is:

                            α = arccos {d/(d+2δ)}= arccos {1/(1+2∆)}                             (3)

Let’s find out how the deviation η on distance S is related to α (fig. 3):

Fig. 3.

It is seen that

η = S∙tg(α)                                                                                                         (4)

Inserting equations (1) and (2) for α to (3), one will get according to known trigonometric formulae and assuming ∆<<1:

- for cylindrical projectile:

or

                                                                                                                                          (5a)

- for spherical projectile:

As we can see, deviation is proportional to ∆ in first case , and to square root of ∆ in latter one. As ∆<<1, it is already clear that deviation for the spherical projectile is much larger. 

Now let’s do some quantitative estimations:

Suggest, we have 8 mm caliber projectile and gap of 50 microns ( in real amateur coilguns the gap makes 100 microns and more because of many reasons, the main of them - low rigidity and roughness of thin-wall  tubes which the barrels are made of). Then ∆ = 0,00625 and shooting at 10 m we shall have maximum scattering of 1,58 m for spherical projectile!