So as I understand it: If you hit a wall at twice the speed you hit four times as hard?  I often ask people this and everyone says "at twice the speed you hit twice as hard".   $K=\frac{1}{2}mv^2$ is very counter intuitive. I have read an explanation that seems to say gravity is acceleration and that's where the squared comes from.  Do we have anyway of knowing if the kinetic energy of an object outside of a gravity field is linear or quadratic? Could the gravity field somehow match the acceleration to give the $v^2$  Bonus question: If you get pulled over for speeding should you ever ask the policeman why kinetic energy is quadratic instead of linear?  $K=\frac{1}{2}mv^2=$($300) dollar fine where I live.
As  says:     [...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.   Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?  My intuition is wrong here, please help it out!
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The question is:     When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?   The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?  Edit  I should add that $f:\mathbb R\to\mathbb R$ to the question. Consider a set $A$ of $10$ integers, $A = \{a_1,a_2, \ldots ,a_{10}\}$. Prove that there is at least one subset of $A$ whose sum is divisible by $10$. Open maps which are not continuous What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to $\mathbb{C}$ (filled in to be $0$ at $0$). There must be a simpler example, using the usual Euclidean topology, right? Is there any canon evidence that FN-2187 actually worked in sanitation? Finn tells Han that he worked in "Sanitation" on Starkiller.  FINN The flooding tunnels are over that ridge. We'll get in that way. HAN What was your job when you were based here? FINN Sanitation. HAN Sanitation? Then how do you know how to disable the shields? FINN I don't. I'm just here to get Rey   It just occurs to me - the "I was in sanitation" line given to Han seems to be the only actual canon evidence of "Sanitation" job.   But was it the truth"?!?!  Finn is always trying to create a positive impression of himself with the Resistance, from the get-go:   He tells Poe that he's rescuing him because that's the "Right thing to do" (with Poe quickly seeing through the fib: "You need a pilot!") He tells Rey that he's Resistance (she buys that because she's a romantic 19 year old still playing with X-Wing pilot helmet) He tells Han that he's Resistance (that one lasts about a minute - Han was not a romantic 19 year old even at 19 :)   In reality, we know that he was 1% of his class, model stormtrooper cadet, personally groomed for promotion by Captain Phasma (; ).  However, I am pretty sure telling Han "Hey, I was 1% of my class, model stormtrooper cadet, groomed for promotion by Captain Phasma"  isn't high on his list of brand management plans.  Far better to pretend to be a lowly "sanitation" worker than top-flight cadet that he really was.  So, my question is - do we have any direct canon evidence proving that the "sanitation" job was real and not a fib to make himself look better to Han/Resistance?  Knowing where trash compactor was isn't direct evidence, though it is circumstantial one. onClick to get the ID of the clicked button I am currently working on a model which takes two parameters and produces a measurement statistic. Think of it as Z = f(X,Y).   Z is a matrix of my statistics and I am creating a surface plot of it in matlab. Basically, I am looking for a mathematical/analytical way of determining if the surface is smooth, or if it is jagged. Do large values tend to be clustered together or are they dispersed throughout the matrix? - that is my question. Basically, how mixed up are the values of my matrix?  I need to run the model over different parameter sets and I want to be able to analytically determine which one of my surfaces is the smoothest, has the greatest clustering of large values, and ideally, has no negative values.   Any help will be greatly appreciated and please let me know if you need any further information.   Cheers What does **Ens** stand for? Earlier someone was asking about the category "Ens" described in Categories for the Working Mathematician. My question is more basic: What does Ens stand for?  Most of the categories have names that are self explanatory, and maybe this one does too.  But for the life of me I can't think of it, and a Google search was no help. Would NAT be eliminated transformed in ipv6? NAT is a wonderful concept used in ipv4. It is the major reason for the existence on internet, in my concerns. There are no networks in the world that doesn't work without employing NAT.  All I seek to know is that, would NAT be required if after ipv6 is implemented, would NAT be eliminated or be used in a different way(Transformed or Upgraded into a new version with some added abilities to tackle any situations or problems that the network specialists or researchers foresee). No command 'bla' found, did you mean:? So, when a command is not found, by what means is the "did you mean:" list populated? What program finds these alternate commands?  What is the meaning of: "(main), (universe)..."? Can I change which program finds these? User with low reputation shows as having modified thread How can this user , reputation 3, show in  as having "modified 7 hours ago", but on the modified thread, , not show at all?  This user also has two medals, which is somewhat puzzling. Embedding $S_n$ into $A_{n+2}$ I am trying to prove that for all $n$, $S_n$ is isomorphic to a subgroup of $A_{n+2}$.  Say $S_n$ acts on $\{\alpha_1,...,\alpha_n\}$ and $A_{n+2}$ acts on $\{\alpha_1,...,\alpha_n,\alpha_{n+1},\alpha_{n+2}\}$.  Let $\sigma \in S_n$ and let $\phi: S_n \to A_{n+2}$ be given by $\phi(\sigma)=\sigma$ if $\sigma$ is an even permutation, and $\phi(\sigma)=\sigma(\alpha_{n+1}\alpha_{n+2})$ if $\sigma$ is an odd permutation.  This way $\phi(\sigma) \in A_{n+2}$.  Now, let $\sigma, \rho \in S_n$.  Then $\phi(\sigma \rho)=\sigma \rho (\alpha_{n+1}\alpha_{n+2})$.  Also, $\phi(\sigma)\phi(\rho)=\sigma(\alpha_{n+1}\alpha_{n+2})\rho(\alpha_{n+1}\alpha_{n+2})=\sigma\rho$.  So, $\phi(\sigma \rho)$ does not seem to equal $\phi(\sigma)\phi(\rho)$.  But the idea is that we want $\phi(S_n) \cong S_n$, and $(\alpha_{n+1}\alpha_{n+2})$ does not permute any members of $\{\alpha_1,...,\alpha_n\}$, so $\phi(\sigma \rho)$ and $\phi(\sigma)\phi(\rho)$ are equal as permutations on this (sub)set.  This feels a little subtle to me and I'm not sure if my idea is valid or not.  If it is not, can this proof still be saved?  I appreciate any thoughts on this.  Thanks. I have a normal door, and left to its own devices it will simply rest about 20-30 degrees. It won't swing shut on its own, nor stay open on its own. It hangs a bit in the middle. Convenient enough to catch you elbow when you're not looking.  I'd like the door to stay put wherever we leave it.  I've heard idea such as slipping cardboard in to the hinge to rebalance the door, etc. but no guidelines as to the thickness of that cardboard, or which hinge to change to move it in which direction.  I would hope you can appreciate how I'd rather not make this a task of trial and error and continually removing the hinge and putting it back on, etc.  I've also heard something about lightly bending one of the hinge pins, but not sure how I should go about that, or "how much" to bend it.  So, I'm looking for some tips on how to fix this (and other) door. Undefined Symbol ___gxx_personality_v0 on link I've been getting this undefined symbol building with this command line:  $ gcc test.cpp Undefined symbols:   "___gxx_personality_v0", referenced from:   etc...   test.cpp is simple and should build fine.  What is the deal? How can I play mp3 files or videos? Can I install Android on my non-Android device? What do brokers do with bad stock? Difference between "did + verb" and just "verb+ed" This is something I always get confuse with (native language German). As far as I know "did + verb" is something done and finished in the past. But isn't it the same with "verb + ed"? What happens if my ESTA expires on the day of arrival? I'm French and will travel the 14th of June to the US. My ESTA expires the 14th of  June 2013 and I'm wondering if my ESTA will be valid when I'll land in the US on the 14th of June 2013.  Does someone know anything about that? I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to process and understand basic ideas behind most of these. However for general relativity there is this one illustration, which is being used over and over (image from Wikipedia):    I always thought that general relativity gives another way how you can describe gravity. However for this illustration to work, there needs to be another force, pulling the object down (referring to a direction in the attached image). If I put two non-moving objects in the image, what force will pull them together?  So where is my understanding incorrect? Or is general relativity not about explaining gravity and just describes how heavy objects bends spacetime (in that case the analogy is being used not correctly in my opinion)?    UPDATE Thank you for the answers and comments. Namely the XKCD comics is a spot on. I understand that the analogy with bent sheet of fabric pretty bad, but it seems that it can be fixed if you don't bent the fabric, but just distort the drawn grid.   Would you be so kind and answer the second part of the question as well - whether general relativity is explaining gravitational force. To me it seems that it is not (bending of spacetime simply can not affect two non-moving objects). However most of the time it is being presented that it does. What's the Pythonic way to combine two sequences into a dictionary? Is there a more concise way of doing this in Python?:  def toDict(keys, values):   d = dict()   for k,v in zip(keys, values):     d[k] = v    return d