Note that although not a homework per se, this kind of question would be better off in the homework section.
Assume no air. This'll give you a lower limit for the energy involved.
Use the suvat equations to find out the initial velocity necessary to land the projectile at an arbitrary distance, say 100km. The launch angle will be 45° for maximum efficiency.
Once you've got the velocity, plug it into the kinetic energy equation along with the passenger+capsule(whatever it might be; probably a lot) mass to get the energy requirement.
The work needed to accelerate the poor shmoe will be equal to that energy. Use the definition of work, taking a few arbitrarily chosen lengths of the barrel(or rails if it's a railgun) to find the acceleration.
Might be a good moment to ask yourself if surviving this many g's is even remotely possible.
One of the possible solutions would be to immerse a person in a liquid-filled tank, making sure that the liquid fills all the body cavities, to prevent tissue tearing due to the acceleration.
Note that without the atmosphere the capsule will decelerate on impact so much harder than that.
No amount of clever engineering will help the passenger if the capsule itself gets squashed.
With atmosphere, you'll need more energy/barrel length/acceleration to get to the same distance, but the landing might be more manageable(air drag, parachutes).
Shooting the projectile at low angle would need more energy/acceleration to reach the same distance, but the landing could dissipate the horizontal velocity by skidding on the surface.
You all are thinking upside down. Imagine a parabolic shaped tunnel between, say New York and London. Drop your capsule in at one end, gravity will do all the work and it naturally stops in London. You never experience accelerations greater then g and get there in good time. The main technical difficulty is digging a tunnel under the Atlantic Ocean.
An interesting thought. The launch of an aircraft from the deck of an aircraft carrier is something to analyze for a real world application of your musings. This should provide some initial data that may be extrapolated for the purpose of of developing your idea further.
Problem is that people want to go to the Greek Islands and not to Alton Towers for their holidays. People are very attached to g, you know. They don't like very much +/1 around it.
I'm sure you thought of this, but with no additional energy added to the 'capsule' it would not rise to the same elevation at the target destination. Even when not considering the friction; as in the case of a pendulum, it would take additional energy to get the mass to the same elevation as it was at, when it was released. Granted, it requires less energy to work with gravity, as to oppose it, one still would not expect a free ride from it.
Also, when the capsule hit the bottom of the arc, it would have a specific force of gravity, when it began to rise, it would have this force, plus the additional force applied, by the movement of it rising. A force dependent on the rate of change in velocity, it was experiencing, in an upward movement. There's more then one way to increase the force of gravity, besides standing still and not moving. Pilots of high performance aircraft experience 'g loading' while bringing the aircraft into a horizontal turn state. When they descend rapidly and pull up. And the 'Vomit Comet' simulates 'gravity less' states by flying up, and suddenly and effectively, reversing direction in a downward direction. G-loading also occurs in high speed elevators. Walking into a 'skyscrapers' express elevator, and, either ascending, or descending; results in a noticeable, disconcerting, and emphatic impact on the passenger.
You want us to discuss so you can cut-and-paste to do your homework assignment?
Perhaps he is thinking of upstaging the human cannonball, with two human cannonballs or more, at the same time. I also, don't think much of OP editing the whole OP, and replacing it with 'discussion'. I could discuss that to no end. Once I got going. I guess it's telling that OP'er has not participated in the discussion since the OP.
Separate names with a comma.