Q:

Can we accelerate a particle to a very high speed by applying force just once?

Suppose you have a tennis ball in your hand. When one would throw it towards a wall such that it encounter the surface of the wall at almost a right angle, the ball would come back in the same direction. 
Now let us suppose the same scenario but as soon I throw the ball, place a similar wall at my place.
Now the ball would bounce back from the first wall and strike the second, bounce back and strike the first wall and this would continue.
Now suppose there is a very very long swimming pool, filled with a fluid which has particles with very less mass, very less inter particle forces of attraction and coefficient of restitution equal to 1.
Now I jump into it and be near one end of the pool.
Now I push the fluid behind me towards the wall of a pool, and as in the case of a ball, the particles will strike the wall with some force F, with Newton's 3rd law, return with F and hit my back with F. I will get accelerated with F/M( where M is my mass).Again with Newton's 3rd law, my back would also push them with F( where my back is acting as the second wall as in the case of the tennis ball) then they would return to the wall then again push me ,accelerating me, and then I would push them.
Since the pool is very long, I would keep on getting pushed and pushed and so increasing my speed and maybe I will beat sound and since it is so long, get somewhat near light.
I know that the coefficient of restitution of my body is not 1 but we can some what increase it to near one by having some sort of suit.
Is this possible? Are there any flaws ? What are they?

A:

It is possible. What you have just described is "where pressure comes from".
Imagine two walls, with a particle bouncing back and forth between them, pushing one then the other. This is your scenario, it means that the particle is exerting pressure on the walls that are holding it in its box.
The amount of pressure the particle exerts will increase if it is travelling faster (is hotter) or if it is heavier, or if their are more particles. This is where the ideal gas law comes from:
https://en.wikipedia.org/wiki/Ideal_gas_law
So not only is it possible, it is how gas pressure works.
Problems with your set up:
However your speeds will be limited. The fundamental reason for this is simple: "How quickly are those magic water particles you hit moving?". Once you are going as fast as them the ones behind you will never catch up to push on you again.
Another (important) problem is that you need nothing in front of you (otherwise pressure in front will pick up to push you back).
So a refined version of your set up might look like this: We are in a perfect vacuum have a perfectly flat plate, with a perfectly flat wall behind it. We put some particles that bounce (perfectly) back and fourth between the two. The two will accelerate apart relatively hard as first, but as the wall-plate distance increases each particle will be used less and less often (longer distance to cover), and will hit the (moving) plate less hard as their velocity difference will be less. Ontop of this diminishing is the speed limit set by the speed of the particles, and the fact that no real materials will have a coefficient of restitution of 1 exactly.
In gas-terms, the external pressure is zero (ideal vacuum was assumed), and the internal pressure (providing force) falls as the volume between the plate and wall increases. Eventually the force pushing the two apart is minuscule, although it never quite reaches zero in the idealised case.

