Physics of a Railgun I have a couple of questions about how railguns actually work, and the mathematics behind them. I understand that the projectile is driven forward by the Lorentz force caused by high currents traveling through the projectile. Because of these high currents, when the projectile initially comes in contact with the rails it can be welded into place instead of pushed forward, and thus needs an initial velocity. 
So...


*

*How do you calculate what the initial velocity needs to be in order to counteract the high current that wants to hold the projectile in place?

*Would there be any difference in the Lorentz force in a pure-translational velocity versus a rotational and translational velocity?

*How high do the currents need to be compared to the mass of the projectile? 

*Where would I find these kinds of formulas for these calculations? 
Thanks!
 A: In a rail gun the projectile is the short circuit.  So whatever your drive through the rails has to be low enough current that it doesn't completely ionize your projectile.

How do you calculate what the initial velocity needs to be in order to 
  counteract the high current that wants to hold the projectile in place?

Acceleration drops off quickly with low currents and at a certain point drag becomes higher than accelerating force and the projectile becomes welded by the resistive heating that occurs. At the same time however, a very high current will cause dramatic rail erosion and resistive losses.
For a high acceleration to occur high currents are required. And this requires a high voltage so that circuit impedance can be overcome. Tradeoffs are made where higher voltages bring higher currents but at the cost of a higher rail separation distance (to avoid arcing). A typical design utilizes around 4 - 10kV, with higher voltages being used at higher energies.
The model is then a balance between your induced acceleration and the drag/adhesion of the type of projectile you are using.
Depending on your design goals, you might have to experiment some to determine an estimation.  A lot depends on your current supply source, materials, and projectile.  The faster you inject the projectile, the less rail erosion and longer life you'll get out of the system.  Sometimes an inductive element needs to be added to lengthen the discharge cycle and reduce the instantaneous discharge that causes vaporization of the projectile and rail damage.
Force is $\vec{F}=i\vec{L}X\vec{B}$
If some work is done to convert the $B$ vector to a scalar product related to L, called L' the acceleration can be estimated.  The magnetic ($B$) vector is very difficult to accurately calculate analytically, as it is based on the rail current, cross-section, length, position of the projectile.  However, a characterization of the geometry can be made as a magnetic field factor, L’. The magnetic field factor is also the inductance gradient of the railgun geometry and is expressed in Henrys per meter.
Then the general the formula for acceleration becomes:
$a=\frac{L'*i^2}{2m}$
Where m is mass and L' is the characterized magnetic field factor.

Would there be any difference in the Lorentz force in a 
  pure-translational velocity versus a rotational and translational
  velocity?

I don't think so.  Drag is probably a bigger factor than whether the conducting projectile rotated or slid. 

How high do the currents need to be compared to the mass of the projectile?

This mostly depends on the acceleration available from your system which is not a linear function due to supply current and magnetic gradient so it is hard to estimate 1:1.  This also depends on the speed you wish to achieve.

Where would I find these kinds of formulas for these calculations?

"Design and Construction of a 1m rail gun" is a good starting point as well as "Design, Fabrication and Testing of an EM railgun".
There are many more out there.
