I have heard that the Voyager spacecrafts received gravitational boosts from some of the planets before they left the solar system.  How is this achieved?  Since we have to conserve energy it seems that whatever speed is gained due to gravity would then be lost as the object flys away from the planet.   Specifically I want to know how we can willingly slow the planets orbit to steal some of its energy.  How do we know that the fly by is going to take energy from the planet rather than lose energy to the planet?
I read in  in this site that the KE a free-falling ball acquires is not originated by the attracting body but that energy was actually stored in the ball when it had been lifted to the height it dropped from.  In this way, it was said, gravity is subject to the conservation of energy principle and cannot change the total energy of an object.  Consider now the maneuver known as a  (also gravity assist) used by space probes such as the . A space probe approaches a planet with velocity $v$, slingshots around, and ends up with velocity $v+2U$, where $U$ is the velocity of the planet.  Consider the energy of the probe. Before, it was $E_i=\frac{1}{2} mv^2$ and after it is $E_f=\frac{1}{2} m(v+2U)^2$. It looks like $E_f$ is much bigger than $E_i$ - but where did the additional energy come from?   Is this not a violation of the conservation of energy principle?
Is there an easy way to attach source in Eclipse? I'm a big fan of the way Visual Studio will give you the comment documentation / parameter names when completing code that you have written and ALSO code that you are referencing (various libraries/assemblies).  Is there an easy way to get inline javadoc/parameter names in Eclipse when doing code complete or hovering over methods? Via plugin? Via some setting? It's extremely annoying to use a lot of libraries (as happens often in Java) and then have to go to the website or local javadoc location to lookup information when you have it in the source jars right there! I am carrying out research to look at differences in fish density and fish species richness when using two different underwater visual census methods. My data was originally count data but then typically this is changed to fish density but I have still decided to use a Poisson GLM, which I hope is right.   model1 &lt;- glm(g_den ~ method + site + depth, poisson)   My 3 predictor variables are method, site and depth which I ordered as factors when I input them.   My response variables are grouper species richness, grouper density and the same for other fish groups. I am aware that density is not an integer and it is numerical data e.g 1.34849. I am now however getting this error:      In dpois(y, mu, log = TRUE) : non-integer x = 0.037500   I've been reading up and many people suggest using an offset, is this the most advisable thing to do? On proving that a finitely generated group has a finite number of subgroups with index $n$. Theorem: If $G$ is a finitely generated group, then it has a finite (maybe zero) number of subgroups of index $n$ for any $n\in \mathbb{N}$.   is a sketch of a proof. It consists of assuming such a subgroup $H$ exists, and allowing $G$ to act on $G/H$ by left multiplication. Such action defines a homomorphism $$\phi :G \rightarrow \text{Sym}(G/H)\cong \mathbb{S}_n$$  If $G=\langle g_1, \ldots ,g_r\rangle$, then the homomorphism is uniquely determined by where it sends the generators $g_i$. In particular, there are at most $(n!)^r$ such homomorphisms.  The last step of the proof requires one to notice that a different subgroup $H'$ of index $n$ in $G$ would produce a different homomorphism, and here is where I'm struggling. I've been able to show that $\text{ker}(\phi )=\text{core}(H)$, yet haven't been able to rule out the possibility of $\text{core}(H)=\text{core}(H')$. If tenured staff are virtually unsackable, why is the drive to find funding so strong? A term for words that change pronunciation with part of speech I'm talking about words like:   construct: CON-struct(n.), cun-STRUCT(v.) present: PRE-sent(n.), pre-SENT(v.) record: RE-cord(n.), ri-CORD(v.)   They are pronounced differently based on whether they are a noun or a verb.  Here are the terms I DON'T think would apply:   Homograph: same spelling, different meanings: for example, bark (the sound of a dog) and bark (the skin of a tree). Homophone: same pronunciation, different meaning: for example, to and two Homonym: same pronunciation and spelling, different meanings: for example: bank (river bank or savings bank) Heteronym: same spelling,  different pronunciations and meanings: for example bow (the front of a ship) and bow (a ranged weapon)   Bow is particularly interesting. It is a heteronym based on the two nouns I described. But in the sense of the body movement, the noun and the verb are pronounced alike (rhyming with wow), thus, not belonging to the category I've explained.  In fact none of the terms explained  satisfy my requirement.  I'm looking for the term for words that have same (or related at best, NOT different) meanings, same spelling and different pronunciation.  Specifically, different pronunciation depending on the part of speech. As against the words which have the same pronunciation in different parts of speech like bill, turn, case (or bigger ones like manoeuvre)  TL;DR:  Fill in the blank:     The noun and verb forms of construct are pronounced differently   because the word construct is a _____   or      The noun and verb forms of construct are pronounced differently   because they are _____s. Idempotent linear operators are projections Heisenberg Uncertainty Principle scientific proof states that:    $$\sigma(x)\sigma(  p_x )\ge \frac {\hbar}{2}.$$  What is the scientific proof of this principle? Difference between Inheritance and Composition How to shuffle all songs in iOS 8.4 music app In the old version of the music app you could shuffle all songs when songs were selected:      This feature seems to be absent in iOS 8.4 and the new music app:      What are the options for shuffling all my songs in the new version of the music app under iOS 8.4? Is the 64-Bit version of Ubuntu only compatible with AMD CPUs? I was told that computers with more than 2 gig's memory need a 64 bit operating system to utilize all RAM.  Is the 64bit Ubuntu download really JUST for AMD processors? I am asking because the disk image I downloaded says AMD64.  So will my new Intel 2.3Ghz Core i3 Dual Core processor work with 64 bit Ubuntu?  It runs the 64bit version of Windows without any qualm. Aristotle's Categories with academic commentary I have often heard it recommended that a user account should be disabled by setting its shell to /bin/false. But, on my existing Linux systems, I see that a great number of existing accounts (all of them service accounts) have a shell of /sbin/nologin instead.  I see from the man page that /sbin/nologin prints a message to the user saying the account is disabled, and then exits. Presumably /bin/false would not print anything.  I also see that /sbin/nologin is listed in /etc/shells, while /bin/false is not.  The man page says that FTP will disable access for users with a shell not listed in /etc/shells and implies that other programs may do the same. Does that mean that somebody could FTP in with an account that has /sbin/nologin as its shell?  What is the difference here? Which one of these should I use to disable a user account, and in what circumstances? What other effects does a listing in /etc/shells have? How to find the joint PDF? Ley X,Y two random variables with density $f(x,y)=8xy \  \text{if} \   0&lt;x&lt;y&lt;1$  Find the joint distribution F(x,y).  I find that $F(x,y)=2x^2y^2-x^4\ \text{  if}  \ 0&lt;x&lt;y&lt;1$ but I dont know how to find the didtribution in other points. Cyclic Group Generators of Order $n$ How many generators does a cyclic group of order $n$ have? I know that a cyclic group can be generated by just one element while using the operation of the group.  I am having trouble coming up with the generators of a group of order $n$.  Any help would be great!  Thanks! If $ad$ and $bc$ are odd and even, respectively, then prove that $ax^3+bx^2+cx+d$ has an irrational root. $\displaystyle ax^3+bx^2+cx+d$ is a polynomial with integer coefficients. It is given that $ad,\,bc$ are odd and even respectively. Then prove that not all roots of the polynomial are rational.  It is easy to see that none of the roots are integer. But how to tackle the rational case? Any help is appreciated. Parsing JSON with Unix tools When is it appropriate to end a question without a question mark? Check if convergent in the end points of the convergence interval I have to find the radius of convergence for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. I think it is $r^{-1}=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{ \frac{n+1}{n+2} }{ \frac{n}{n+1}}=1$ so r=1. Then I have to find out if $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$ is convergent in  the end points of the convergence interval. I think I have to check for $|r|$. But we can't use ratio test (L=1) or integral test(not decreasing) for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}$. How can I then check if it's convergent in the end points of the convergence interval? Does the regularization of a divergent infinite sum yield a unique value?  I.e. do different regularization schemes acting on the same infinite sum produce the same exact value independent of the regulator?  What, exactly, do these values mean?  Or what are they?  My understanding is that they are not "convergent" values.  (Sorry in advance for being a physicist. Capture console exit C# I have a console application that contains quite a lot of threads.  There are threads that monitor certain conditions and terminate the program if they are true.  This termination can happen at any time.  I need an event that can be triggered when the program is closing so that I can cleanup all of the other threads and close all file handles and connections properly.  I'm not sure if there is one already built into the .NET framework, so I'm asking before I write my own.  I was wondering if there was an event along the lines of:  MyConsoleProgram.OnExit += CleanupBeforeExit;