Let's look at a situation inspired by Galileo's example. An observer on a train moving smoothly with velocity v either drops a ball or throws it horizontally with speed -v, ie with speed v in the opposite direction. We can compare this with an observer on the platform who either drops a ball, or who throws it (forwards) with v. We show the two points of view. Galileo chose lead or rock as the material so that the weight of the projectile would be much greater than the forces exerted on it by air. In our animation, air resistance is neglected.
What happens on the platform. From our viewpoint, also at rest in the station, we see that the ball dropped by the man at right on the platform falls vertically. The ball thrown horizontally at v by the man at left on the platform follows a parabolic path.
What happens on the train. The train is travelling at v, so when the man at left on the train drops a ball, it follows a parabolic path: if follows the same path (with respect to us) as the ball thrown by the man on the left of the platform. However, the man on the train sees the ball land at his feet: in his frame of reference, the ball he dropped has fallen vertically.
For symmetry, the man at right on the train throws a ball at -v with respect to the train. We see its horizontal speed as v+(-v) = 0: in our frame of reference, it falls vertically, but to him it travels along a parabola.
Of course, anything is possible in an animation. Science relies on experiment and observation. Next time you are on a train, stand still and drop a small dense object (a coin would do) on the floor. Does it land near your feet, or does it land where your feet were when you released it? Depending on how tall you are and how you hold it, it will take about half a second to fall. If the train travels at say 100 k.p.h., your feet will have travelled about 15 metres during the fall, so the results of the experiment are likely to be clear. (For safety reasons, do not drop or throw objects outside a train. However, you and a friend might like to try experiments using a bicycle.)
You might also care to conduct an experiment when travelling in an aircraft. Sitting in your seat, you drop a fork into your lap. You and the fork are both travelling at several hundred kilometres per hour, with respect to the ground. It takes the fork a quarter of a second to fall the first 30 cm. During this time, the plane (and you, and the fork) travel 40 metres. I suggest doing this experiment only because I am quite confident that the fork will fall in your lap: it will not collide with your chest at hundreds of kilometres per hour. For this experiment, only the relative motion is important.
Galileo's work reported in Dialogue Concerning the Two Chief World Systems actually makes a lot of progress towards what we now know as Newtonian mechanics, which is our next topic.
