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An electron can travel faster than light, but we have no means to do it because it doesn't respond anymore to our electric fields that are the only way that we know to make him move.
As an analogy, a tomate that is not a charged object, doesn't respond to electric fields.
We can put a tomate in a super maximum hyper hadron collider and the tomato will not move.
The conclussion will be that the maximum speed that a tomato can achieve is zero and it has an infinite mass. Nothing can travel faster than a tomato that doesn't move at all.

The speed of light is critical in three equations, the Lorentz transformations.  They're the ones that show that, as you go faster time seems to slow, mass increases, and length decreases.

These have all been experimentally demonstrated.  In fact, GPS satellites have to take this into account because their clocks tick at a slightly different than does a clock on the surface of the planet: it's not enough for a human to notice in the everyday world (just as the slight increase in mass when you run compared to standing still isn't big enough to matter given the forces involved), but if you need to calibrate highly accurate timepieces, it does.  The increase in mass, and time slowing down, has also been observed in particle accelerators.  A particle that decays in a certain amount of time takes longer to decay if it's moving faster, and one moving faster hits with more mass than one would expect if you didn't take relativistic mass increase into effect.

Common in all three equations is the Lorentz factor, 1/sqrt(1-[v^2/c^2]), where v is your velocity and c is the speed of light.  The way it works in the mass equation is:

Mass of moving object = mass of object when "unmoving"  times Lorentz factor

Let's say that you are travelling at the speed of light, v=c.  Then v^2/c^2 is 1/1, or 1.  So now you have to find the square root of 1-1, or 0.  The square root of zero is zero.  But now the factor is 1/0, or infinity.

So at light speed, the mass of the moving object is its rest mass...time infinity.  So if the object has any mass, any at all, it would have infinite mass at light speed.  That's why we know that photons that move at light speed can't have any mass to start with, and why you can't accelerate anything with mass to light speed: it would achieve infinite mass (that is, it would outweight the universe).

Here's the other thing: in order to accelerate something, you need energy.  The energy to accelerate something is directly proportional to the mass of the something.  The more mass, the more energy needed.  Well, to accelerate an object of infinite mass, you therefore need infinite energy. 

Incidentally, the reason why your tomato example in the particle accelerator makes no sense is because there isn't enough energy in the particle acccelerator to move a tomato.  There's a lot of energy in there relative to the particles and some individual atoms that are being moved, but no where near enough to significantly effect the huge number of atoms and particles in a tomato.  Your example is about as silly as arguing that because I can't push the space shuttle into orbit, therefore there's no amount of energy that can push the space shuttle into orbit, which is obviously a dumb thing to say.  All it's saying is that a single person doesn't have the energy, not that the energy isn't available.

Ah, but if that space shuttle, or that tomato, is moving really, really fast, then we run into the energy problem.  At 99.9% of lightspeed, the tomato masses 22.366 times as much.  At 99.99%, 70 times as much; at 99.999%, 223 times as much; at 99.9999%, 707 times as much, and so on until it reaches infinity at 100%.

So, in order to accelerate something with mass to equal lightspeed, we need more energy than exists in the entire universe, while moving something that weighs more than the entire universe.  Since that obviously makes no sense, you can't accelerate a tomato to exactly lightspeed.  And if you can't get it to lightspeed, you can't accelerate it it faster.