What is the maximum speed , we can get without dangerous mass increase ?

yeah, mass increases. the lorentz factor is: 1/sqrt(1-v^2/c^2) where sqrt is square root, v is the velocity of the object, c is the speed of light, and ^2 means squared. you would plug in the velicity you wish to find out about to v. time decreases for you by this factor, mass increases, and length decreases.(ive had to derive these formulas from simple mechanics formulas; they're correct.) If you would plug in velocities to this formula, you would notice that the effect is negligable for any velocities not very very close to the speed of light. Well, we definately can't go the speed of light because if you look at the formula, then v^2 would be equal to c^2 so v^2/C^2 would be 1, and 1-1 is 0. this would mean that the denominator would be 0 which is impossible. we also can't go faster because then v^2/c^2 would be greater than one, thus subtracting that from one would yield a negative answer which would then be under a square root sign. this is also impossible. that's how we know we can't go as fast or faster than light speed. Now, as far as how fast we can actually go, that depends a lot on biology. as we get faster and faster, our mass increases and our length(and therefore volume) decreases. now, the formula for density is m/V(mass over volume,) so our bodies would get more and more dense the faster we went. this is why there would be a limit and we couldn't just go so close to the speed of light so that time would slow down for us but not for everyone else, thus basically traveling forward in time. The question of how fast we can go becomes a question of how dense our bodies can become before either the pressure builds up too much in our cells, or our blood gets so dense it can't properly carry oxygen to our cells. The thing is, you really have to be very close to the speed of light to feel this effect. the speed of light is 299792458 m/s. that's close to 300 million meters per second. now lets say we go 200 million meters per second. that's pretty darn fast, and i dont even know if we could go that fast due to the enormous amount of energy it would require to produce such accelerations. but lets say we do. plugging this into the formula, we get 1.16. we multiply this by the original mass, giving an increase in mass of only 16%. this isn't really all that much, we could probably handle that increase in mass, but i really don't know, my expertise is not in biology. Still though, 200 million meters per second is pretty damn fast, anything much less than that would most certainly be negligable. you can try out some speeds for yourself to see what kind of effect it would have(you need a graphing or good scientific calculator otherwise it will round to 1 in certain places and you won't get any change at all.) We won't have to worry about them coming anywhere close to speeds that affect our mass anytime soon though.

I think you may be confusing virtual mass with actual mass. The mass of an object does not increase with velocity, but when approaching light speed there are some unusual effects. Distances and time contract, these are called Lorentz(sp) contractions. For any amount of mass, to accelerate the mass to light speed would theoretically take infinite energy. Since there is no such thing, we'll have to be satisfied with sub light speed, or find new technologies.

I think you may be confusing virtual mass with actual mass. The mass of an object does not increase with velocity, but when approaching light speed there are some unusual effects. Distances and time contract, these are called Lorentz(sp) contractions. For any amount of mass, to accelerate the mass to light speed would theoretically take infinite energy. Since there is no such thing, we'll have to be satisfied with sub light speed, or find new technologies.