Experimentally verifying some facts What are some thought experiment/real life experiment to convince me these two facts, so that I can test them actively:
Fact-1: a. The speed of a falling body is non constant, and is increasing. b. Also, two bodies with different masses falls at the same speed.
(Note that a. I don't own this room b. I don't have photoelectric (or whatever sensor they use to measure interruption in a beam of light to measure speed) sensor :) 
How do I be sure of these facts ? For the first part, I think I need some very very big height to test this, but I can't attain that much height. For the second part, I am having difficulty testing that as almost everything doesn't work in the long run (i.e there's a minisucle, but noticable with naked eye, difference in their positions when two objects with different mass falls). 
Fact-2: A person sitting in a train moving with constant velocity can't figure out (when the windows are shut) if the train is moving or staying still. 
I don't think I can test it because I can perfectly tell, when being inside a train, when the train is moving or not, simply because the speed is always nonconstant, but how I can be sure what happens in the ideal case ?
 A: Fact 1: Note that the atmosphere makes this sort of experiment inaccurate; on Earth, different falling objects are slowed to different degrees by air resistance, and they also reach terminal velocity. I suppose you could use a vacuum chamber.
As for part a, it should be something that you can see under regular circumstances (e.g., watching a dense object fall to the ground from a high window). If you are willing to generalize free fall to a condition of having a constant net force (that is, you're willing to accept that forces can be similarly applied by means of gravitational attraction and physical application) then I would suggest an experiment in the spirit of the treatment of Newton's laws in An Introduction to Mechanics by Kleppner and Kolenkow; I think this sort of experiment is common in physics education. They describe using a track that eliminates friction, but that isn't necessary for this experiment; as long as you apply a constant force greater than friction, you are applying a constant net force. Basically, attach a rubber band to an object, and stretch it horizontally to some length, keeping it at that length as it moves (thus applying a constant force). Note that there will be a bit of a hiccup as the switch from static to kinetic friction occurs. You can use a small spring scale if you want to be more careful. In any case, you will observe that the mass moves with a constant acceleration.
Fact 2: This is tricky because on trains, we infer that we are moving if we hear/feel noise/vibrations consistent with our past experiences of movement, even if we are not accelerating. In my experience, this is true to a lesser extent with airplanes, and even less with planets.
A: Fact 1 :- Well actually, If I understood you correctly, you meant to say that how can we confidently say that two objects fall with the same acceleration regardless of their masses, without taking into concern the small inaccuracies caused due to the limitation of the Speed of Light and of our naked eye.
Remember, Einstein's Theory of Relativity may work here. In order to verify fact 1, you can take a feather and a large metallic ball, in a vacuum chamber. Take the two objects at a certain height and using highly accurate computing systems and mechanisms you can drop those two objects from that height (of course there will be little of inaccuracies and non-simultaneity in this process, which I will ellaborate further in the conclusion) therefore the two objects will fall with the same acceleration and will reach the surface at the same time(again inaccuracies caused here because of the low agility of the observer and the propagation of light at a constant speed. But anyways with this experiment we can easily verify the first fact.
Conclusion for Fact 1 :- We can therefore conclude that, there is always some inaccuracy left, and that we can never be 100% efficient. But still in this case even if the objects fell at the same time for us, even if there might be any inaccuracy, that inaccuracy might have affected all our other observations and that we might imply that the Fact is true and verified, but let us say another creature with a more accurate and agile mind than ours observes the same event, he can easily point our inaccuracy and can state that the two objects do not land at the same time, but then we will insist and say that the objects landed at the same time. In order to solve the dispute, what Einstein did was reformulated the Theory of Relativity (Relativity literally meaning relation with one another) what he did was that he made everything relative, time, length, mass etc. He invented some equations which told both the observers to compromise between their measurements equally, so as the event will be the same both of them. To our surprise, I applying his concepts in philosophical aspects too and therefore we conclude that this fact will be true for us and will not affect anything until another creature comes with much more accuracy. 
Fact 2 :- The second fact too is based on the Theory of Relativity. In case of practical situations you can differentiate easily whether the train is in motion or not because of the variable velocity of the train and many other external factors. In case of ideal situations where a train might go on a track smoothly with a uniform velocity and that we shut the windows completely, closing ourselves from the outside world. In this case think of a separate Universe of your own, because now you are detached from the outside world. If you try and open one of your windows and if there is a train outside moving exactly at the same velocity as yours, you will observe that the train appears stationary. If you ignore or close the view of your surroundings outside and just observe the train that is moving with a velocity equal to that of yours' train, again you cannot differentiate whether you both, the second train and yours train are in motion or not.
Conclusion for Fact 2:- If you are moving with a completely constant velocity, like in the thought experiment I stated in my explanation above, you cannot differentiate whether you are moving or not, if another body is moving with the same relative velocity, that body too will be added to your world of non-differentiation in your state of motion. So for you both, time will pass equally, your length will be the same, but if another body comes with a different velocity than yours, you will observe that for that body time is passing more slowly as compared to yours. This is a whole new branch known as the Special Theory of Relativity which is very interesting to explore. Hence, the fact is verified.
A: In both cases your question is really "How can we be sure that the Laws of Physics are correct?" 
What we observe in everyday life often contradicts what the Laws of Physics say. Feathers and cannonballs do not fall at the same speed. Thrown objects do not continue moving in a straight line at constant speed, they move in arcs, slow down and stop. The Laws of Physics describe what happens in ideal situations. Everyday life is full of non-ideal situations, each of which is complicated by a variety of factors. Physics generalises these observations into the Laws by ignoring irrelevant factors which can or could be eliminated if careful arrangements are made. 
You seem to be asking how you can make ideal observations. You cannot. You can only make the situation you are observing closer to the ideal. That is what scientists do in laboratory experiments. They go to a lot of trouble to eliminate as far as possible all of the complicating factors. For those which cannot be eliminated, they examine what happens as the complicating factor is gradually reduced, then they extrapolate to the case when the factor is absent.
So for example with the cannonball and feather they remove the air, performing the experiment in a vacuum. With the train, they continuously measure its speed and supply a little more or a little less power when it is going too slow or too fast. The latter is called a feedback and control system. 
A: How can we confirm that velocities of airborne objects change?
Throw a ball upwards. If everything which was not supported by matter had the same downward velocity, it would be impossible to throw a ball up in the first place. Instead you will notice that the ball slows down, comes to a stop, then starts to speed up relative to you. Furthermore the change in velocity appears continuous rather than instantaneous, as the ball appears to have a certain "hang time" near the top of its trajectory. The assumption that the velocity is totally constant on the upward half of the trajectory also raises a very interesting question, which is why the ball does not continue off to infinity -- how does it "know" to stop when it does? Wheras if the velocity is just steadily decreasing, that velocity is able to carry some information about "how far have I gone" directly and we do not need an extra explanation for this fact.
Similarly, drop a glass cup a distance of one centimeter and see if it breaks. Now consider dropping it a distance of twenty meters. If its velocity impacting the ground were the same either way, how would it know that it has fallen farther and is supposed to break? 
Clearly we would need a theory where there is some notion of "momentum" and this increases with the time the object is falling: but your understanding of the momentum of an object would have to decouple from velocity and have nothing to do with it. And we could test that, too: show me the fly which hits you with a small velocity but so much momentum that you get knocked ten feet into a wall, and then show me the car which hits you at 120 mph but does not impart any momentum whatsoever. If momentum really has nothing to do at all with velocity then both of these should be possible.
How can we confirm that different masses feel the same gravitational acceleration?
Fill two identical plastic water bottles, one a quarter-full with water, the other full with water. Observe that one is approximately four times heavier than the other. Now drop them side-by-side and observe that they both hit the ground at the same time, which means that they must have had identical $v(t)$ curves -- rather than what you may have expected, that the heavier one hits the ground in a quarter or maybe a third of the time that the full one does. (The little differences become irrelevant when you can push the mass difference to very large discrepancies and you do not see very large discrepancies in the time that they hit the ground.)
There's actually an even better thought experiment: imagine dropping three identical balls side-by-side but just barely not touching; they must all hit the ground at the same time because they are three identical objects. Now use a piece of double-sided tape to stick together two of these balls so that they now constitute an object of mass $2M$. Do you really expect that this little piece of sticky tape is somehow able to make these two balls fall twice as fast as the other one? Or do you expect the sticky tape to be mostly irrelevant?
How can we confirm that reference frame velocity does not affect kinematics?
Start juggling while facing North, then turn to face East, then turn to face South, then turn to face West, juggling all the way. If we purely consider the rotation of the Earth alone and ignore its orbital motion and the motion of the Solar System about the Milky Way and so on, then you just transitioned from moving a thousand kilometers per hour to your right, to moving a thousand kilometers per hour straight ahead, to moving a thousand kilometers per hour to your left, to moving a thousand kilometers per hour backwards. Surprisingly you did not need to modify your juggling strategy to accommodate these massive speed changes, did you?
Well maybe the Earth "drags space along" with it in its rotation, or something: there is an absolute space and you just happen to not be moving relative to it when you stand on Earth. How lucky! But you can also juggle in a train with the exact same approach, indicating that there is no first-order effect from the speed of the train on your juggling, unless absolute space is also so flimsy that the train wins out over the Earth. Compare this with the things that we think you definitely could observe: train sees a car stuck up ahead on the tracks and immediately pulls its brake as hard as possible. Given what you know about how people lurch about in this context, do you think you could still juggle by the same methods you learned on land? I'd daresay not.
Of course you can also repeat this experiment on a plane, for even greater velocities and even less observed "noise" in the constant velocity -- but people do get much more skittish when they see people doing "weird" things on planes rather than on trains, so please be careful.
A word of caution
Physics, like all science, has two levels. There is the "theory" level where we figure out some rules that help us phrase the models, and the "model" level underneath it where we try to construct a representation of reality.
In general we build these models within these theories, the theory tells us how the model behaves, and we discard models aggressively until we find models which compare favorably with experiment.
You may have noticed that the experiments above do not disprove very strange theories, but rather that the models of those theories get aggressively more complicated. And I want to say that this is true and it's endemic -- there is no getting rid of it. If you are searching to prove that Newtonian mechanics is more correct than other adjacent theories, you will find that this is not strictly-speaking possible. Those other theories can also model reality very well. However their models for reality are really complicated, in order to explain the same results.
