If someone fired a gun while standing on a train that could go as fast as a bullet, would the bullet just fall down, travel as per normal, or travel at the total speed of the bullet + the train?

It would travel at the speed of the train plus the usual speed of the bullet. To people on the train, it would look simply normal -- the bullet would appear to travel at only normal bullet speed. To people on the platform, the bullet would appear to be moving faster than normal bullet speed -- the speed of the train + the normal bullet speed. So basically it would be boring, with no cool visual weirdness. The only potentially cool thing would be if the shooter stood on the platform and fired the bullet. If the bullet traveled at exactly the same speed as the train, the passengers on the train would see the bullet hovering beside the train.

The starting velocity of the shooter is relevant. If the gun is travelling at bullet speed with the train, then the bullet will come out at bullet speed relative to the train. If the gun were fired backward from a train travelling this fast, then it might be bullet speed relative to an observer on the train, but stationary (subject to gravity) relative to an observer on the side.

Assumptions: Train is travelling at a speed at which bullet will travel and the person who travels bullet is on train CASE 1: As seen by person who shot the bullet Since the person is standing on train so the speed of train, relatively to him will be zero, therefore the speed at which bullet will travel, as seen by the person who fires, will be its natural bullet speed. CASE 2: As seen by person standing on ground For such person, firstly if train is travelling with speed 'x' and also bullet travels at its speed which is again x, then for him the bullet will travel at a speed 2x! Hope you understand. It's simple concept of relativity!

A person on the train will see it traveling at bullet speed. Another on the ground will see it traveling at (bullet+train) speed. And no, it will not fall down.

Well, if you fire a 'Bullet' from a 'Bullet Train'(let's call it so for now) it will follow a pretty complex trajectory. I don't think, you can give a name to the path followed by the bullet. I will try to get you to visualize how the bullet will behave in such a scenario. Suppose you fire the bullet at right angles from the train. Let's assume the direction the train is travelling towards as the 'x' direction. 'y' would then indicate the direction of the bullet. Now the bullet is being fired from a moving frame of reference with respect to the ground. Give the velocity of the train as 'v' which is the same as the velocity of the bullet. Converting the moving frame to a stationary one, we have a bullet being fired at a net velocity of '(Root of 2)v' i.e 1.414v at a 45deg. angle with x. This simplifies the problem by assuming that you have fired the bullet at 1.414v velocity, at an angle of 45deg. with the x direction from a stationary position. From the top view, the trajectory will more or less be a straight line(Air resistance will do cause a deviation, but such a deviation would NOT be noticeable to the naked eye). Now for the yz plane, the bullet will follow a parabolic trajectory-similar to the trajectory that a stone will follow if you throw it parallel to the ground, except that this is at an extremely high velocity. Again, this parabolic trajectory is a pretty elongated one owing to the speed of the bullet(Snipers do take this effect into account when aiming at targets over long distances).This would again NOT be discernible with the naked eye. In conclusion, when combining both the frames, the bullet will seemingly follow a pretty complex trajectory, very close to a straight line. The minor deviation would not be discernible with the naked eye. YES, the bullet will curl, BUT, the effect would not be as dramatic, as you expect it to be.

You have inertial, the bullet has inertia, the gun firing it has inertia. The only difference from normal is a change in atmospheric resistance since the air surrounding the outside of the train isn't moving at the same speed. So yes, the bullet would fire and move as normal but compensation for extreme "wind" may be required.

It depends on which refrence frame you are watching.If you are watching from refrence frame of train the bullet will have a normal speed and if you are watching from earth's reference frame velocity of bullet will be twice velocity of (Velocity of train will also add up).

It will move forward. Obviously the speed of your bullet is faster than a train. Using the concept of relative velocity, the bullet will move forward but with a velocity less than what it would have been had you shot it being stationary. The velocity of the bullet with respect to you would be the bullet's actual velocity with respect to a stationary observer - the velocity of the train Taking the case when the train is moving faster that the speed of bullet, The bullet will move backwards towards you.

In classical physics, it well travel as fast as the total speed of the bullet + the train, but in relativity speed aren't summable because of the cosmic speed limit (speed of light in the vacuum). It well be something like this: considering V0 is the speed of the bullet. and V1 is the speed of the train. the speed of the bullet in a moving train will V2=V1+V0. in relativity V2 = (V1+V0)/(1+V1*V0/C^2) and C is the speed of light