# 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?

• In your pool example, you can generate a wave that travels with some velocity $v$. Once you're also moving at $v$, that wave will not accelerate you. Your speed can't increase by coming into contact with something moving at the same speed or slower than you. Your notion that the wave always imparts a force $F$ to you is incorrect. Sep 12, 2019 at 17:28
• Wait, why is your mass changing spontaneously? You're just floating in a pool. Sep 13, 2019 at 13:08
• Oh sorry, I actually misunderstood your post pardon me. I will be deleting my comment thanks for sharing your view. Sep 14, 2019 at 4:20

In your pool example, you can imagine the pool and the waves you've created in it to be like a battery. The energy you pushed the water with becomes the energy of the wave (which is a combination of kinetic and potential energy).

Assuming you could remove losses, the wave would continue indefinitely.

But as soon as you try to "surf" on it, you are removing the energy. As the wave accelerates you forward, you are pushing the wave backward. This pushing removes energy and makes the wave weaker. At some point, the wave would disappear and can no longer accelerate you. Your "battery" is empty.

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.

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.

• Good point about pressure, but your answer "it is possible" doesn't match with the OP's question. Pressure doesn't spontaneously increase without bound in a sealed container, just like you can't be infinitely accelerated by an oscillating wave. What the OP describes is not possible. Sep 12, 2019 at 17:31
• I agree their are a great many issues with the exact situation the OP describes, but something in the same spirit of it can indeed work. If you have a gun say the air molecules going back and forth between the back of the bullet and the back of the barrel is what drives it forwards like his swimmer. This (could) recycle the same air molecules many times. The pressure falls as the bullet (swimmer) goes forwards and the volume behind them rises. I see nothing in the OP's question that leads to "pressure spontaneously increasing without limit".
– Dast
Sep 12, 2019 at 17:38
• I will add some of the "real world" flaws to my answer though.
– Dast
Sep 12, 2019 at 17:40
• The way I read the pool example, it suggests that you can generate a single wave that bounces between the ends of the pool, and ride that wave to increase your speed enough to "get near light" speed. I suppose that's true if you had an enormous wave with enough power to do that in the first place, but you can't keep recycling the wave's energy to accelerate continuously. In your gun example, the pressure will equalize eventually and the bullet will stop accelerating. It's trivial to accelerate to enormous speeds by imparting enormous force, but that doesn't seem to be the OP's scenario. Sep 12, 2019 at 17:49
• I agree Nuclear Wang. I have added something very much like the gun, but in vacuum to try and keep the "pressure from outside" from mattering.
– Dast
Sep 12, 2019 at 17:52

Well, your thinking is right but you are not considering the energy loss of the tennis ball due to striking the wall nor the damping effect of the atmosphere. The tennis will lose a lot of energy due to deformation of the tennis ball when hitting a wall.

Likewise with the long swimming pool. The resistance exerted on the swimmer by the water is even stronger than the air resistance of the tennis ball. This can be realized by considering the difference in the velocity of runners versus swimmers. Part of that difference is due to the resistance of the water. If you have ever tried to run in a swimming pool you realize this effect. Also, the water bouncing off the wall will lose energy due to effects of fluid dynamics on the drops hitting the wall.

When you push water behind you if will diverge and hit the wall at angles not parallel to you and the wall. Hence just a small fraction of the water will bounce back towards you. Much of the water will momentarily stick to the wall and run down the wall. These processes will greatly decrease the amount of water striking you. The water will also lose energy by bouncing off your back due to the small deformations to wherever the water strikes your body.

• Right but it's not water. It's some kind of an ideal fuel. There is resistance mainly due to the inertia of the particles and inter particle forces, and 8 have already written that the particles of the fluid have very less mass(so less inertia) and very less inter particle forces, so there won't be much resistance due to the fluid. Sep 14, 2019 at 4:23