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Works like a slingshot, the slingshot is encased in the ship and the released projectile imparts its momentum onto a pusher plate attached to the ship, moving it forward. The projectile is then picked up again and it repeats.

There is no exhaust and the fuel is reused endlessly as the fuel is a free moving object with momentum imparted to it using electricity, take the nautilus shell where using electrical energy the spiral acts as a chamber to accelerate a projectile in a circular motion, the projectile free floating, not connected to the ship, at some amount of momentum the projectile goes from circular motion to linear motion where it strikes the ship imparting its force and moving the ship forward. The projectile is then sent back to the chamber to endow it again with momentum, and the process repeats.

Imagine a man on a special sea-saw, he walks to the furthest edge of one side of the sea-saw, the sea-saw does remain horizontal as it requires a certain amount of force with tip to one side, the man then jumps up and down on the furthest side of the sea-saw providing enough force causing the sea-saw to tip downwards. The external force is gravity in this example, while in the ship the external force is momentum endowed in a projectile.

The LHC uses electrical power to accelerate particles, and the slingshot launch system also uses electricity to propel a projectile. In this case, the accelerator is encased inside the ship, and the projectile being inside the ship but not part of the ship.

The more elegant design has a nuclear power plant that magnetizes the fuel and turns it into a state that is not a particle such as plasma, the fuel is contained in a magnetic wall and the fuel in momentum is directed magnetically so it does not damage the material parts of the ship, the plasma is magnetized opposite of the pusher plate and magnetically repels the fuel pushing the ship forward. As the plasma loses momentum, it is again directed to the circular chamber to have momentum endowed to it.

A projectile in motion imparting its force onto another projectile implicates Newton's Third Law of Motion: Action and Reaction. A well-balanced system would cause the first projectile to lose all its momentum and then reintroduced into the circular motion chamber, a well-made system would pulse the propulsion in an engine design probably something resembling a handbag, where the bag storage is the circular momentum building chamber and the handle of the handbag is projectile heading towards the pusher plate and the pusher plate being at the apex of the handbag handle where the force is imparted to the ship. The handbag storage area is perpendicular to the handle, so it is more like a flat pot. Electromagnets are employed and magnetism propels the ship. A well-made engine design would pulse the ship forward continuously.

It violates Newton's third law of motion?

Take billiards or snooker. Newton's third law tells us that during the collision the forces between the two balls are equal in magnitude and opposite in direction. However, this doesn’t mean that the white ball must always travel in the exact opposite direction to the black ball. Here’s why:

1. Force vs. Motion: the third law applies to the forces during the short collision, not directly to the final directions of motion. The outcome (i.e., the velocities and directions) depends on the angle of impact, the spin on the balls, friction from the table, and the timing of the force application.
2. Collision Geometry: In a head-on collision between two identical balls, if struck perfectly, the cue (white) ball can transfer most of its momentum to the object (black) ball, potentially stopping or even following it. In many real shots, the white ball isn't deflected backward because the collision isn't perfectly "mirror-like" – instead, both balls can continue moving in similar directions if struck with the right angle.
3. Impulse and Momentum: Momentum is a vector quantity. While the forces during the collision are equal and opposite at every moment, the net impulse (force applied over time) can impart a momentum to the white ball in the same direction as the black ball if the contact is such that their paths are aligned. This is why, in some shots, the white ball can "follow" the black ball into the pocket.

In summary, Newton’s third law governs the interaction forces, but the final motion results from how those forces are applied in time and space. The white ball following the black ball into the pocket losing the game is entirely consistent with physics when you take into account collision geometry, impulse, and friction.

It violates Conservation of Momentum?

This is where people get confused. They say that it cannot work because for instance, firing a gun in a building is conservation of momentum and the bullet will not go though the walls of the building. For the bullet to penetrate the wall of the building you must be standing outside the building. They do not understand the design we are proposing here. The condition is one projectile striking another object to impart movement to the second object, a car bumping another car for instance causing it to move forward and bump the next car. The confusing bit is that the same simple condition is encased within the ship, which brings up conservation of momentum. In cased does not mean part of the ship. Take the example of a gun floating in thin air without gravity in a tube with one end closed off. Once the gun is fired, the bullet reaches the end of the tube, the bullet imparts its force onto the end of the tube and the tube achieves forward momentum. The gun and the projectile are not part of a closed system. Conservation of momentum requires the projectile to exert force onto the ship. What about brakes and orthogonal directions, if you use an external brake, guide, or some other mechanism that diverts the gun’s recoil to the vertical direction, then you are effectively removing the horizontal recoil component from the gun. This means the horizontal momentum imparted to the bullet (and then delivered to the tube when it strikes the closed end) wouldn't be balanced by an equal and opposite horizontal recoil. As a result, the tube would gain net horizontal momentum.

There is no exhaust, so it cannot work?

Gravity slingshots have no exhaust and that works, so exhaust is not essential for motion or momentum.