When experimenting with minecarts with chests for a storage and selection system to direct the carts to specified areas I discovered they stacked into each other even with full inventories.
Learning this I decided to try personal transportation using regular minecarts. When carts collide with each other they repel each other so I thought maybe this will be the method of motion. Turned out I was correct.
My method for this was to stack 20 or more carts into a one blocks space, then break the rail they are on and nudge them ahead via a piston pushing the stack into an anvil or chest. Once done that I placed a boat above the minecarts in such a way that when the boat falls it hits the edge of the stack of carts and they shoot away in a random direction,(first removing the anvil or chest and the block that it was on so the boat can fall into only the minecarts). It was common for 1 or 2 minecarts to fall away from the group but the group never lost integrity over all.
Now that the carts are cornered somewhere, (a player can be in the stack of carts when they are launched by the boat), they can be interacted with by a player. Get in a cart and hit the W key while facing into the block that the carts stopped against, this will move you in the direction away from the block in front of you. If you aim diagonally and hit W you may go diagonally, (easier to do when in a corner).
The minecarts never lose momentum. Each time a player gets out and someone else gets in the speed remains. The stack of carts should keep their formation so long as no one or no mobs interact with it via physical contact against the side of the stack.
This makes a drive able cart in minecraft without the need of rails.