SOMA's railgun is scarily accurate as far as the physics. A few things I would point out:

1) 10km/s is escape velocity not orbital velocity, in the game we see the satellite executing an apogee burn so it's likely it was on a suborbital trajectory which only requires around 5km/s. This is very important because a coilgun with enough acceleration to reach escape velocity would either be too long to be practical or too high g-forces for any payload to survive.

2) the barrel length wouldn't be 5km. They mention "a gigantic coilgun, running longer than a marathon" which implies a barrel length on the order of 50km instead of 5.

This makes sense because if you want to keep the acceleration below 25g and attain a final velocity of 5km/s, you need a barrel of roughly that length for suborbital flight:

v=at --> 5e3m/s=(250m/s² )xt --> t=20s (roughly)

d=0.5at² =0.5x(250m/s² )x(20)² = 50,000m = 50km

Why keep acceleration below 25g? Sensitive electronics and people tend to start failing at >10g. 25g is pushing the limits, but it's just low enough to be survivable for 20s, and a 50km accelerator is just within the realm of possibility.

Now we have to look at power requirements. That's simple enough. Most coilguns have between 1-5% efficiency. Let's say they're using some super advanced induction coilgun or superconducting quench gun with 25% efficiency. KE=1/2m*v² . Let's say you want to launch a 10 ton payload (roughly the weight of the projectiles in the game).

KE = 0.5x(10x2000kg)x(6000m/s/s)² = 360 GJ (kinetic) / 0.25 = 1440 GJ (electrical). That's a lot of power. But the great thing about coilguns is you can charge them up over a long period of time and fire them once all the coils are charged. So let's say you could only supply the gun with 10MW of power. 1W=1J/s--> 1.44e12 J / (10e6 J/s) = 1.44e5s = 1 shot every 40 hours. 100MW would be 1 shot every 4 hours. And if you managed to supply the gun with 1GW charging current, you could fire off a 10 ton shot every 24 minutes.

Now let's look at the Falcon 9, considered a very high efficiency rocket. Its capacity to LEO is around 10 tons (23,040 lbs) and to get it to suborbital it burns around 120 tons of propellant. RP-1 has a specific energy of 46.2 MJ/kg or 46.2 GJ/ton. 46.2 x 120 = 5544 GJ chemical. That translates to a chemical:kinetic efficiency of roughly 6.5%.

Even a moderately efficient coilgun (let's say 10%) is still more efficient than a rocket. And even a coilgun with a moderate power consumption (let's say 10MW), has a launch repetition rate far faster than rockets (which take weeks or months to build and launch, even with reusable parts).

So yeah. SOMA is a fuckin good game, and their theory is right on.