If a space ship accelerated constantly, would its astronauts constantly feel the forward movement? I know that if a space ship suddenly traveled very fast, its astronauts would be fly against the back wall, potentially getting hurt. If the space ship suddenly stopped, they would also fly against the front wall.
Suppose that a space ship could accelerate at a constant rate, e.g.:
Time    | Speed
------------------
1:00 pm | stopped
1:10 pm | 100 Kph
1:20 pm | 200 Kph
1:30 pm | 300 Kph



*

*As the space ship is accelerating at a constant rate, would the astronauts constantly feel the forward motion of the craft for the duration of their journey?

*Would the astronauts begin to feel that the motion is becoming faster and faster?

*If the space ship continued to accelerate in this manner for many hours, would the astronauts eventually be injured by the great speed?

 A: Suppose the spaceship was accelerating constantly at 1g, what would that feel like? Well Einstein gave us the answer to that: it would feel exactly like standing on the surface of the Earth where the acceleration due to gravity is 1g. This is (one statement of) Einstein's equivalence principle. If the acceleration were continued for many hours that wouldn't be any different to being on the Earth's surface for many hours, which of course we all do.
A: I'll try to give as simple an answer as possible to each of your three questions.


*

*The astronauts will always feel the acceleration (but as Muphrid pointed out, in this case it's tiny and might not even be noticed)

*It is not possible to feel speed while in a spacecraft. Astronauts in orbit travel at 28000 km/h but feel absolutely nothing, even if they're outside. Similarly, inside a car you do not feel the speed, only the change in speed (i.e. acceleration - and note that acceleration can be in any direction: forward, backward, left, right, up or down). You feel the speed only if traveling through the air, where you feel the air dragging at you.

*Speed does not cause any harm at all, as you never feel any of its effects.


Now a couple of provisos.
First, if your speed became a large fraction of the speed of light, any particles (even single atoms or nuclear particles) you encounter would fly clean through the spacecraft and through the astronaut. If you're going fast enough, these particles in effect become nuclear radiation, and that can indeed harm you.
Also, it is not possible, even in theory, to keep on increasing your speed by 100 km/h every 10 minutes. You will never be able to go faster than 300,000 km/s even if you keep on accelerating forever. The same force will give you smaller and smaller acceleration as you go faster. This is what Einstein discovered with his theory of Special Relativity.
A: You're describing constant acceleration, and this would result in the astronauts feeling a constant force in the direction of travel exerted on them by the spacecraft.
This particular acceleration is actually quite small--about $.005g$--and as a result, it might hardly be noticed, let alone be dangerous.
A: The overall speed doesn't matter, and the astronauts would have no way of directly perceiving their velocity, but yes they would constantly perceive the acceleration.  The effect of acceleration would not accumulate over time---i.e. if it wasn't harmful at one point in time, it wouldn't be harmful later.  This is neglecting longer-term biological effects like re-distribution of blood, etc.
A: The situation is a very interesting one. The answers given are nice answers, but I would like to see what would happen if the force (acceleration) kept acting, 'pushing' the space-ship for ‘infinitely’ long time. 
The acceleration of the space-ship ‘forward’ would be equivalent to a gravitational field acting backwards. This is according to the principle of equivalence and the law of inertia.  Had the acceleration been $g = 9.8ms^{-2}$ the astronauts would feel as if  they were in their room, at their home, here on earth. 
As long as the acceleration is maintained the astronauts would have this experience, and the speed of the space-ship would be constantly increasing.  But I don’t think this would be without consequences. I am saying this because the mass of the space-ship would be increasing, adding to its inertia. Assuming the power (force) remained fixed, then the space-ship would tend to the speed of light, but never actually reaching it. This is because its acceleration would be decreasing up to a point where it would become zero, which would happen when the speed of the space-ship reaches the speed of light.  It resembles the situation of the terminal velocity of an object falling through the air, but the physics is different. Other more interesting effects could take place.
