Remember the neat little experiment whose result may be counter intuitive to some of those who embrace the “all bodies fall the same way inside earth’s gravity” ‘doctrine’ without quite understanding it? Here it is again - if the video should not come up fast enough, here it is on the MIT website.
It is a simple set-up and great idea for a project involving children. Older ones may be able to calculate the required lengths of the wooden angle's arms and the required position of the cup.
I challenged readers to come up with a short and philosophical reason for why the end of the arm reaches the table faster than the free falling ball although much of the force onto the wooden plank just presses on the hinge without being able to accelerate it.
‘Philosophical’ means: Get to the vital core and convince with fundamental insight – no formulas.
The being faster is necessary for the ball to separate from the golf tee and land in the cup. But the cup is not important – it is a gimmick much like the guy wearing a lab coat. Important is a deeper understanding of the physics involved:
In old ages long ago, people thought that a feather does not fall as fast as a cannon ball. Nowadays, people know that the feather only falls slower due to air resistance. This in turn has become somewhat of a 'doctrine' in the sense that many who 'know' it do actually not really understand the underlying reasons and thus apply the doctrine where it does not apply. They are surprised by the experiment in the linked video, because the wooden plank’s tip experiences a downwards acceleration larger than if in free fall! (Look at the golf tee where it supports the ball: They separate!)
I would like to thank all the readers for their willingness to participate and stick their neck out. I learned a lot from the comments. Especially: I saw how widespread this misconception is that mathematics is necessary to understand deeper connections. People seem to be ready to be duped into accepting something as correct because it is full of formulas. It goes so far that even those who are well on the way to correct answers do not trust their own abilities merely because they have not added equations.
Most important lesson today: Mathematics is a language. You use it to communicate content. If the content is not correct, the fact that it is expressed in form of formulas only makes it worse!
I often experienced students or colleagues telling me something impossible and when I immediately object they are annoyed if I don’t even look at their calculation. Why should I? It is wrong. Why spend an hour trying to figure out where the formulas are not applicable? Life is too short.
Do never trust people who try to tell you that you just do not understand their argument because your mathematics isn’t good enough. It is actually only people who do not understand their own field properly who ever do that.
Anyways – John Marretada came closest. He did not have the confidence to go all the way, but he had three important ideas.
I do like his considering the mass to be close to the hinge (1). Whatever the length of the plank, whatever the starting angle (as long as it is below 90 degrees), if all the mass is close enough to the hinge, the massless tip can reach any velocity, even light velocity and beyond.
Another fact holding true regardless of lengths, cups, masses, whatever, is that if all the mass is in the plank’s tip and the starting angle is 0 degree, the ball’s acceleration and the tip’s initial acceleration will be exactly identical (2) (because both are then free falling bodies, at least for an infinitesimal short time). This second one is all that John was missing.
I like his total avoidance of equations. What I did not like about his argumentation is that it too soon and too often mentions aspects that are (a) not quite equal or should be different neglecting this or that and (b) dependent on going a certain fraction (of length or angle) here or there. Even if it is all true, such is never as convincing as stuff that is obviously (a) identical especially if it is so moreover in (b) extreme situations. That’s why (1) and (2) above are didactically superior:
(1) In the extreme limit of all mass at the hinge, the tip's acceleration is identical to infinity (which is yet another extreme that overshadows all doubts about something maybe not being quite enough to do the trick).
(2) In the extreme position of all mass being in the tip and the starting angle being 0 degree, the accelerations are identical.
John’s further idea is also very good, namely considering to take some of the mass from the tip into the plank. In other words, ‘smear’ the mass along the plank. All mass away from the tip also ‘wants’ to fall as freely as the tip, too, but cannot because it is connected to the tip, and so it must drag the tip along. Thus, the tip obtains additional acceleration. Given the extreme position (2) implies that the accelerations are identical already. Hence, the additional acceleration makes the tip’s acceleration larger than that of the ball. Now we have already proven the main point: the tip can be faster than in free fall.
Somebody else may prefer a more mathematical "the smearing of the mass leads the acceleration monotonically from extreme limit (2) into the direction of extreme limit (1)" without mentioning physical forces at all. I consider this a matter of taste and both versions should be pointed out.
Again: We have proven that the homogeneously-smeared-out-mass plank is faster. Only problem left is that our starting angle is still zero degrees! Note: the plank’s acceleration is identical to zero at the extreme of 90 degrees and increasing the starting angle approaches that extreme monotonically: So if you want to prove that the tip is faster also at a finite angle above zero degrees, you need to make convincingly sure that the added acceleration from smearing the mass along the plank is not overcompensated by the slowing down due to the starting angle being larger than zero degrees.
So again: Only problem left is that our starting angle is still zero degrees! But we can change the angle because the tip’s acceleration is larger than the ball's acceleration by a finite (i.e. not just infinitesimal) amount. Therefore, we must increase the starting angle by a finite amount if we wanted to compensate for the additional acceleration. So let us compensate in this way, increasing the starting angle until the acceleration of ball and tip are identical again (the ball is of course always experiencing the same acceleration regardless of where it starts).
Now we are basically finished because after compensating, we have a finite angle at which the tip and ball have identical accelerations (always try to stick to identical if at all possible). Take half that compensating starting angle instead and the tip will be again accelerated more than the ball but at a finite angle (> zero degrees). That is it: QED.
How large is that angle? Is it close to 35 degrees? Where does the cup go? I do not care. It is not important for answering the question posed. Those details probably need calculations with formulas, but that is exactly why they are not interesting.
UPDATE: The short version was provided by Colin in his comment below.