Ball dropped from train A ball is dropped of a train traveling with high velocity, to a man standing near the  track, the ball........
A) falls down vertically
B) moves horizontally
C) follows an elliptical path
D) parabolic path
 A: Consider this scenario in vacuum as and when the ball is released it obeys newtons 1st law and hence it remains in rest.
If we neglect the effect of air due to the speed of train, the ball falls down vertically since it experiences only the gravitational pull.
By considering the effect of air and if the ball is close to the train it may come along the train (for some time), then it may slowly go down
The movement of the ball is with respect to ground
A: Consider the forces on the ball when it is released.
If we neglect air resistance, then only the force of gravity will be acting on the ball, and it will accelerate vertically from the point of view of the observer on the train. It will retain its horizontal velocity it had already because it was moving with the train. This is a parabolic path from the point of view of the man on the ground.
If air resistance is not negligible, then there will be a drag force in the direction opposite to the velocity, while gravity still acts. This will tend to reduce the horizontal component to zero, while maintaining a vertical component. The form of the motion will depend on several things, but will most resemble elliptical from the point of view of the observer on the ground because the horizontal component of velocity will decay.
A: To a man standing at the bank, the ball will  appear as thrown horizontally (with initial velocity as the velocity of the Train )  , therefore it should follow a path of a projectile thrown horizontally under the action of gravity . With the above  info.  one can get to the nature of path easily.
A: Assuming that we can ignore air friction, and that the velocity of the train is constant (neither was specified) then
a) From the point of view (in the reference frame) of the person who dropped the ball, it moves vertically.
Now look at things from the point of view/reference frame of the person who catches the ball.
b) Assuming that the gravity gradient of the earth can be ignored (which was not stated) and that the velocity of the train is small compared to the orbital velocity of the earth at that altitude (which was also not stated) the ball will follow a parabolic trajectory.
c) Assuming that the gravity gradient cannot be ignored, then depending on the velocity of the train, the ball will follow either an elliptical, a circular or a hyperbolic path. See Newton's Cannon.
d) If air friction cannot be ignored, none of the choices is correct, in either frame of reference.
EDIT - Oh yes, and in the frame of reference of the ball, of course, no motion (by the ball) occurs.
