A woman on the rear platform of a train moving with constant velocity drops a coin while leaning over the rail. Describe the path of the coin as seen by (a) the woman on the train, (b) a person standing on the ground near the track, and (c) a person in a second train moving in the opposite direction to the first train on a parallel track

(a) To the woman on train, the coin would appear to fall down in a straight line. The horizontal component of velocity of the women in the train and the coin are same, and cancels out relatively. Therefore, only the vertical velocity of the coin can be used to predict its perceived motion.
(b) To the person standing on ground, the coin has both a net horizontal component and a vertical velocity component. The coin accelerates toward the ground while maintaining a constant speed in the direction of motion of train.The situation of the coin is similar to an object in projectile motion, thrown from some height with some initial horizontal speed. Therefore, the path traced by the coin would resemble a parabola.
(c) To the person in the train running parallel and in the opposite direction, the perceived horizontal component of velocity of the coin would be the sum of the magnitude of velocity of both the trains while the coin would still appear to fall with the same acceleration towards the ground.
Therefore, the coin would still appear to trace a parabola but of larger range.