I assume calculations start with the payload weight. A massive foundation would be needed lots of concrete and rebar meters thick. Also a harmonically sealed tunnel a thousand miles long filled with helium at a low pressure,very expensive. Steam or a linear motor powerful enough to achieve the crazy speeds needed to hurl such a huge object in to low orbit and a heat shield. I envision the spaceship a mile long and pointy like a big dart filled with 400 astronauts.
Yes, you start with the payload weight and the acceleration. That gives you an amount of energy you need to transfer, which tells you how many power stations you need to build.
Then you need to consider what happens when your ship leaves the gun barrel: it slams into the atmosphere at 10 km/s. This has 2 consequences:
- the ship will start to decelerate at a high rate. This rate is important, you need to have 8 km/s left when you're outside the atmosphere. And you need this rate to be survivable (so less than 5G or so), which will dictate the aerodynamics of your ship.
- the friction will heat the ship rapidly. This means you need a heat shield, which you have to add to the payload weight, and you need to start the calculation again.
there's a load of related questions, some of which have lots of calculations to inform your design.