Your calculations are on the right way, but not quite correct.
I'm a physician and a friend of mine and I did some math recently about railguns and their physics.
Let me do a couple of calculations for you.
First I'll calculate the average acceleration of the projectile while still in the barrel (between the rails). We assume that it reaches 10000 m/s
muzzle velocity with 10m barrel length:
a t = 10000 m/s (equation number 1)
0.5 a t^2=10 m (equation number 2)
a being the acceleration in m/s^2 and t the time in seconds.
Solving these two equations for a and t, this gives us:
a=5000000 m/s^2 and t=0.002 s
The acceleration force is the force that must be applied to the projectile in order to accelerate it, and at the same time, this is the force the
projectile has to withstand. It is:
F=m a ; with m being the mass of the projectile
Assuming we have a projectile with a mass of 0.01 kg (10grams), the force is:
F= 0.01 kg x 5000000 m/s^2 = 50000 N
That means in terms of g-forces (a somewhat misleading term), that the projectile has to withstand 509684 G's.
And this is how came to the result:
The gravitational force on the projectile would normally be:
Fg = m g ; where g is the gravitational acceleration on earth(9.81 m/s^2)
Fg= 0.01 x 9.81 = 0.0981 N
The number of G's is simply how many times the gravity is acting upon the object, hence:
F / Fg = 509684
Let's move on to recoil, the most interesting topic.
Actio = Reactio means if you are applying a force on something, it "pushes back". When a glass of water is standing on the table, the glass pushes
against the table with the amount of its gravitational force and with the same force, the table is pushing against the glass. If this isn't the case,
the glass of water falls through the table.
With the railgun, it's the same. If the railgun is pushing against the projectile, thus accelerating it, the projectile is pushing against the
railgun. It doesn't matter how the force is generated.
Let me elaborate a bit. The net momentum of the system must remain the same. At the beginning, before the gun is fired, the momentum is zero, since
nothing is moving. At the end, the momentum will be zero again, because momentum can't be created. The projectile is moving in one direction, the gun
in the opposite, both having exactly the same momentum. One momentum is negative (because of its opposite direction), hence when you add up the two,
you get zero.
The formula for the momentum is:
P = m v ; m being the mass of the object and v the velocity
The momentum of the projectile would be:
p = 0.01 kg x 10000 m/s = 100 Ns
Let's assume that the gun has a mass of 100 kg.
The momentum of the gun has to be the same as the projectile's momentum.
If you solve the above equation for v, you get the velocity of the gun and shooter after the shot:
v = p / mGun = 1 m/s
This means that a gun weighing 100 kg will move at 3.6 km/h after having shot a tiny projectile weighing 0.01 kg at 36000 km/h. This requires a force
oof 500N to stop the recoiling barrel within 10cm (just as a sidenote).
Feel free to play with the equations and try different combinations of barrel lenght, muzzle velocity, projectile weight etc. You will get interesting
That took me quite a long time to write! I hope everything is clear and helps clarify the physics involved a bit better. In order to find out
whether the projectile will handle the acceleration, you'll have to look up the tensile strength of different materials. And this topic gets even
nastier when you take into account the friction, air drag etc.!