There's probably a better place to ask this question, but I don't know what it is. That being said...
Which will go further if a batter manages to hit it with a baseball bat: a baseball thrown to the batter at 90 miles per hour or one thrown at 60 miles per hour?
The one which goes at 90 mph. For details see this subcomment.
Now I wonder what it is like to be a bat.
Meta-comment: I know that the question is trivial and this isn't a physics discussion board, but still am surprised by the number of downvotes. I am interested in the voters' reasons. Is it OK to ask for an advice concerning diet, headaches or car purchase, but not physics? Why is that?
Notice: The post has been at -5 at one moment, the comment reflects that state. If its long-term karma is positive or zero, the implicit hypothesis in my question - i.e. that people are downvoting questions about physics more than questions about headaches - may not be true.
"Car purchase" got votes because its author mumbled things about biases and rationality in it. The dieting and headache prevention got votes because the community recognises them as "rationality topics", topics which have been used as examples where majority wisdom is wrong and rationality can help. I suspect the Alicorn's post got more votes because people saw that it was written by Alicorn.
Physics is not a recognised rationality topic, and so the above post got voted down. Had it been about quantum physics it would have received upvotes.
N.B. I had downvoted all three of your examples as well as the above post. I also downvote all the AI articles.
I also downvote all the AI articles.
Good luck with that, you are risking running out of your downvote quota. (I have upvoted the comment to provide you some more fuel.)
Had it been about quantum physics it would have received upvotes.
This is now a testable hypothesis.
I didn't downvote, but might have if there hadn't already been enough downvotes when I arrived.
The relevant difference, for me, is that this question is not just trivial but is in a well-settled field. Classical mechanics is an idealized, fully specified model. There's no special reason to apply rationality to it or to ask LWers. Even a very difficult mechanics question would be off topic on LW and better placed on a physics forum, IMO.
In contrast, diet and medicine are two famously unsettled and contentious fields and LW posters constantly use them as examples of topics humanity needs more/better info on, examples of failures of the mainstream science process, etc. After LWers tell me a randomly chosen dietologist is unlikely to help, it makes sense to ask on LW about diet.
The car purchase question is at least about human behavior, so answers might have pointed out biases / irrationality in proposed solutions. I think it should have been posted in Discussion, though.
I propose this guideline: if your post is a question and comments provide answers, how likely is it that future readers will benefit from having this content on the site? In my view, much more probable with the diet question than with the mechanics questions.
There is little evidence that LW can do better on factual questions than mainstream medicine or dietology. I believe that there are few posters who are significantly better than an average doctor, but there are also others who are worse and the readers have not much data to decide who is which. Given that the LW advisors have little or no chance to examine the patient the way normal doctors can and thus are basing their diagnoses solely on verbal communication with the patient, the most rational advice would be "go and see your doctor". Since there are a lot of different answers under that post, I wonder whether the future readers would benefit from that at all, rather than being harmed.
More generally, learning well settled fields is in many respects preferable to pondering difficult unsolved problems. At least you get a definite answer.
There is little evidence that LW can do better on factual questions than mainstream medicine or dietology.
To collect evidence (whichever way it might point) we should encourage factual questions on these topics. I'm interested in such evidence because some LWers have indeed claimed that a rational approach such as a good LWer might be capable of should indeed do significantly better than mainstream experts.
Since there are a lot of different answers under that post, I wonder whether the future readers would benefit from that at all, rather than being harmed.
It's not clear one way or another, but they might benefit, e.g. from being exposed to certain opinions and references. Certainly, to extract concrete advice and follow it, they have to judge the many options discussed themselves, as well as look for other options not mentioned. That's what LW rationalist training is supposed to be all about :)
More generally, learning well settled fields is in many respects preferable to pondering difficult unsolved problems.
In real life you don't get to choose which problems to solve. As a basic example, for me to save my own life, I have to answer questions about medicine (and perhaps diet), even if no human has been able to answer them before.
To collect evidence (whichever way it might point) we should encourage factual questions on these topics.
That's true. But the advice asking posts don't do that particularly well. I have no idea how to collect meaningful evidence on LW's expertise from what's been written there.
Meta-comment: I know that the question is trivial and this isn't a physics discussion board, but still am surprised by the number of downvotes. I am interested in the voters' reasons. Is it OK to ask for an advice concerning diet, headaches or car purchase, but not physics? Why is that?
I notice the post has dragged itself back up to the positive. Your comment has something to do with that. :)
Well, if we ignore the force being exerted on the bat by the batter, then we can use the principle of conservation of momentum, and if we make-believe that the collision is perfectly elastic, then we can say that kinetic energy is conserved as well. This gives us two equations to play with:
where the vs are initial velocities, ms are masses, and ws are subsequent (after-collision) velocities. Algebra then lets us derive two equations for the subsequent velocity of the ball in terms of the masses of the bat and ball, the subsequent and initial velocities of the bat, and the initial velocity of the ball:
and
In both these equations [edit: this is wrong; see addendum below], the subsequent velocity of the ball is an increasing function of the initial velocity of the ball (when all the other variables are treated as constants). Clearly, balls that are hit faster will travel farther, so this would seem to suggest that the faster pitch will be hit farther. However, in addition to the simplifying assumptions we made at the beginning, this analysis also assumes that the after-collision velocity of the bat is independent of the speed of the pitch, which is surely not true, so this reasoning cannot be said to have definitively resolved the question.
EDITED TO ADD: Except velocity and momentum are vector quantities, so the initial velocity of the ball should be regarded as having sign opposite of that of the initial velocity of the bat. This would seem to mean that our momentum equation is actually saying that the slower pitch will be hit further ... I'm confused.
This would seem to mean that our momentum equation is actually saying that the slower pitch will be hit further ... I'm confused.
No reason to be confused, remember you have ignored the force exerted by the batter. If the ball is enough heavy and fast, it will simply shoot away the bat and continue forward. (Edit: the mistake you made is
(when all the other variables are treated as constants)
which you can't, since the posterior bat velocity isn't independent of the prior ball velocity.)
Let's make it explicit:
Let w and W are velocities of the ball and the bat after the hit, v and V are the velocities before the hit, m and M are masses of the ball and the bat, respectively. Then we have
v+2MV}{m+M})
This means that if the bat is ligther than the ball, the faster you throw, the slower the ball returns, or it doesn't return at all, if 2MV is lower than (m-M)v. On the other hand, if the bat is heavier than the ball, the faster the ball moves initially, the faster it returns.
Of course, one shouldn't disregard the batter completely. A better model is to assume that part of the batter is co-moving with the bat. Then, M is the aggregate mass of the bat and the movable part of the batter, which is very likely to exceed the mass of the ball, and then, the faster the ball goes, the faster it returns.
Even better model expects that part of the energy is lost on the bat-batter boundary and in the batter's muscles and joints. Now we should have a reasonable model of how much this is. My hunch is that this amount in fact increases (absolutely, not only relatively) with increasing speed of the ball: if the ball is really fast, it will knock the bat out of the batter's hands before the impulse could be transmitted between the batter's body and the bat, making it effectively a free bat and ball system. But I don't have any idea about actual numbers.
By the way, I envy you Americans such a Newtonian sport. It's not so easy to modify those examples to naturally fit soccer or ice hockey.
Edit: I have now found that the bats are a lot heavier than the balls, which makes all speculations about the batter's physiology irrelevant to the question.
Intuition pump: if the masses are equal and one isn't moving, then as per a Newtonian cradle, the moving ball stops completely and the other takes up its motion.
Also, hooray for Maxima!
(%i1) solve([m*v+mm*vv=m*w+mm*ww, m*v*v+mm*vv*vv=m*w*w+mm*ww*ww], [w,ww]);
(%o1) [[w=v,ww=vv],[w=(2*mm*vv+(m-mm)*v)/(mm+m),ww=((mm-m)*vv+2*m*v)/(mm+m)]]
The problem is complicated, so lets simplify and then look at two limiting cases and see what they suggest.
First, it is simpler to ask which ball leaves the bat in a "forward" direction faster. Presumably the one that leaves faster also goes further. The collision is "elastic" neither the bat nor the ball gets permananetly crushed in a way as to take energy out of the system.
The two limiting cases to look at 1) bat is moving at 0 velocit and weighs same as ball Then either the 60 or the 90 mph ball come to a dead stop on contact with the bat and the bat recedes at 60 or 90 mph backwards. 2) the bat is moving at 0 velocity and weighs a tremendous amount more than the ball. Then the (60, 90) mph ball hitting the very heavy stationary "bat" bounce off it in the forward direction at (60, 90) mph and the bat remains essentially stationary.
So we see for bats between the weight of the ball and infinity (and I think this is the true range of bat weights), the 60 mph ball never goes faster than the 90 mph ball, but sometimes the 90 mph ball goes faster. I think details of how fast the bat is actually going will not matter, that we have surrounded the problem with this simplification, and the fast ball goes further when hit well.
Assuming in both cases the ball is struck squarely, the one going 90 mph, though note that the speed of the ball affects the probability that the ball is struck squarely (or at all).
Most likely, the one at 90. The ball bounces, so moving it towards the bat faster makes it faster when it bounces off.
Others have answered in equations; let me see if I can put the answer in words, for possibly easier understanding. When ball hits bat, its kinetic energy is (with some reasonably high efficiency) converted into potential energy by the ball deforming from its equilibrium shape; then that gets reconverted to kinetic energy (now going in the opposite direction) as the ball springs back to roundness. In other words, the ball bounces. (If you assume a rigid ball the problem changes a lot.) So we can rephrase the question: Which will bounce higher, a ball that's just dropped from some given height, or a ball that is thrown downward so it hits the ground at a higher speed? Now we intuitively see that it is the latter. Bouncing from a moving bat is no different from bouncing off the ground.
Now there are some caveats to that; the efficiency of the energy conversion is not constant, it's some function of the ball's speed - at some point the ball doesn't bounce at all, it just disintegrates (or bursts into flames, if the loss to heat is sufficient). But for speeds achievable by human throwing arms, this effect can probably be ignored.
Generic intuition pump: take much more extreme values: what happens if the ball is thrown at a million miles per hour? Maybe it bounces back at a million miles per hour, maybe the batter looks in dismay at the pile of splinters that used to be his bat ... so it depends of what your model of the batter is. The question as asked doesn't allow to answer that question, though you could probably estimate a reasonable model of a batter.
No reason to be confused, remember you have ignored the force exerted by the batter. If the ball is enough heavy and fast, it will simply shoot away the bat and continue forward. (Edit: the mistake you made is
which you can't, since the posterior bat velocity isn't independent of the prior ball velocity.)
Let's make it explicit:
Let w and W are velocities of the ball and the bat after the hit, v and V are the velocities before the hit, m and M are masses of the ball and the bat, respectively. Then we have
This means that if the bat is ligther than the ball, the faster you throw, the slower the ball returns, or it doesn't return at all, if 2MV is lower than (m-M)v. On the other hand, if the bat is heavier than the ball, the faster the ball moves initially, the faster it returns.
Of course, one shouldn't disregard the batter completely. A better model is to assume that part of the batter is co-moving with the bat. Then, M is the aggregate mass of the bat and the movable part of the batter, which is very likely to exceed the mass of the ball, and then, the faster the ball goes, the faster it returns.
Even better model expects that part of the energy is lost on the bat-batter boundary and in the batter's muscles and joints. Now we should have a reasonable model of how much this is. My hunch is that this amount in fact increases (absolutely, not only relatively) with increasing speed of the ball: if the ball is really fast, it will knock the bat out of the batter's hands before the impulse could be transmitted between the batter's body and the bat, making it effectively a free bat and ball system. But I don't have any idea about actual numbers.
By the way, I envy you Americans such a Newtonian sport. It's not so easy to modify those examples to naturally fit soccer or ice hockey.
Edit: I have now found that the bats are a lot heavier than the balls, which makes all speculations about the batter's physiology irrelevant to the question.
Intuition pump: if the masses are equal and one isn't moving, then as per a Newtonian cradle, the moving ball stops completely and the other takes up its motion.
Also, hooray for Maxima!