Please read till the end, before you answer.  I'll start off with the examples to get to the point.   Drop a block from 1 meter and it will have speed v when it reaches the ground, with total work done on it w.   Drop it from 2 meters and it won't have twice the speed, since it has less time to gain speed. Surprisingly,  2w work is done to it.   Another example. I know it's weird but it's important  A 1 Ton block of lead in space is at rest. A human is standing beside the lead block. The human is going to use  chemical potential energy 2000 J to do work on the lead block and move it and displace it  2000 meters,  using a force of 1 N on the block,  Neglecting the force he uses to move himself to be able to push the block. How much time has he spent doing force?   Since force 1 N is done to on a 1 ton(1000 kg) block,  and acceleration is force over mass,  the acceleration is 1/1000. using the distance traveled by  acceleration formula,  the time needed is 2000 seconds. He has increased the momentum of the object from 0 to 2 thousand.   In another scenario,  A human used 2000 J chemical potential energy using force 1 N on a 1 ton block of lead, but this time it was moving with a speed 50 m/s. The amount of time he would need to move it would of be less than 20 seconds, because of the block's speed.  The increase in  momentum of the object must be less than 20. Surprisingly, In both scenarios  the human used chemical potential energy 2000 J, which is absurd, because the one in the first scenario must have used more energy.   Tl;Dr 2 humans consume the same amount of joules but the first uses so much "effort" and the second guy uses no effort but in the end,  same joules consumed.   This is what I'm trying to say. The amount of work you have done is affected significantly by the initial speed  of the object.   What consumes more food(energy) , adding 2, 000 momentum to an object or 20?  Physics tries to tell me they use the same energy?I don't understand how in each  of the scenarios the human  consumed the same amount of Joules  even though he clearly would be so much more tired after pushing the block at rest than the moving block . This absurdity is slowly driving me insane.  Where did I mess up?
Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.
What are $x$, $y$ and $z$ if $\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y} = 4$ and $x$, $y$ and $z$ are whole numbers? What are $x$, $y$ and $z$ if $$\dfrac{x}{y + z} + \dfrac{y}{x + z} + \dfrac{z}{x + y} = 4$$ and $x$, $y$ and $z$ are whole numbers?  MY ATTEMPT  Let $u = x + y + z$.  Then the equation can be rewritten as $$\dfrac{x}{u - x} + \dfrac{y}{u - y} + \dfrac{z}{u - z} = 4$$  Suppose I set $$1 = \dfrac{x}{u - x} = \dfrac{y}{u - y}$$ and $$2 = \dfrac{z}{u - z}.$$  Then I get $$x = y + z$$ $$y = x + z$$ $$z = 2(x + y),$$ so that $$z = 0 = x + y,$$ which is impossible.  Next, suppose I set $$\dfrac{4}{3} = \dfrac{x}{u - x} = \dfrac{y}{u - y} = \dfrac{z}{u - z}.$$  Then I get $$4(u - x) = 3x$$ $$4(u - y) = 3y$$ $$4(u - z) = 3z$$ so that $$12u - 4(x + y + z) = 3(x + y + z)$$ which implies that $$12u = 7(x + y + z) = 7u$$ from which it follows that $$u = 0.$$ This is, again, impossible.  Alas, here is where I get stuck.  Any hint(s) will be appreciated. $z$ is a complex number, by the way.  I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem.  One of the most "satisfactory" things I've done so far is to expand both the real and imaginary parts for the equation \vert z + 1/z \vert^2 = a^2. These parts yield the constraints:  $2xy(1+1/(x^2+y^2)^2)=0$ ;   $x^2-y^2+\frac{2(x^2-y^2)}{x^2+y^2}+\frac{(x^2-y^2)}{(x^2+y^2)^2}=a^2 $  I've graphed the equations and supposedly the intersection should be my set of points, but this method is far from analytical.  What can I do? Any hints on how to approach this? Zip all files in directory? I stumbled upon an example, which needed to be compiled with the additional option --shell-escape. Using this option it worked just fine, but I'm curious:     Why or when is this option needed and what does it actually do? How to sort a list of objects based on an attribute of the objects? Is there a quick way to lock my Mac? On Windows you can hit the keys Windows+L to lock your computer if you step away.  I don't want to have to remember if sleep currently asks for a password, if there is a time out where my screen isn't locked during a short period. Even with Lion supporting resume of many apps, I don't want to log out.   I do want to be sure data is secure when stepping out for lunch or a smoke break. Calling a function in main How do I list the default installed packages? I am developing an offline installer for all versions of Ubuntu, and I need Ubuntu's default installed packages list. Is there a way to get this information from any server (web server)? Any script to fetch any Ubuntu version's default installed packages list.  I will give the Ubuntu version, and the script will fetch the packages list.  Note: I need at least a server address. I can write a script for this. Find three Poisson-distributed random variables, pairwise independent but not mutually independent How to show that the determinant of the following $(n\times n)$ matrix  $$\begin{pmatrix} 5 &amp; 2 &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 2 &amp; 5 &amp; 2 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 2 &amp; 5 &amp; 2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; 2 &amp; 5 &amp; 2 \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; 2 &amp; 5 \end{pmatrix}$$  is equal to $\frac13(4^{n+1}-1)$?    More generally:     How does one compute the determinant of the following  (where the three diagonals are constant)?      $$M_n(a,b,c) = \begin{pmatrix} a &amp; b &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ c &amp; a &amp; b &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; c &amp; a &amp; b &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots&amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; c &amp; a &amp; b \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; c &amp; a \end{pmatrix}$$      Here $a,b,c$ can be taken to be real numbers, or complex numbers.   In other words, the matrix $M_n(a,b,c) = (m_{ij})_{1 \le i,j \le n}$ is such that $$m_{ij} = \begin{cases} a &amp; i = j, \\ b &amp; i = j - 1, \\ c &amp; i = j + 1, \\ 0 &amp; \text{otherwise.} \end{cases}$$  There does not seem to be an easy pattern to use induction: the matrix is not a diagonal  of the type $M = \bigl(\begin{smallmatrix} A &amp; C \\ 0 &amp; B \end{smallmatrix}\bigr)$ (where we could use $\det(M) = \det(A) \det(B)$ for the induction step), and there are no lines or columns with only one nonzero entry, so  gets complicated quickly.  Is there a general pattern that one could use? Or is the answer only known on a case-by-case basis? It's possible to compute the determinant by hand for small $n$:  $$\begin{align} \det(M_1(a,b,c)) &amp; = \begin{vmatrix} a \end{vmatrix} = a  \\ \det(M_2(a,b,c)) &amp; = \begin{vmatrix} a &amp; b \\ c &amp; a \end{vmatrix} = a^2 - bc \\ \det(M_3(a,b,c)) &amp; = \begin{vmatrix} a &amp; b &amp; 0 \\ c &amp; a &amp; b \\ 0 &amp; c &amp; a \end{vmatrix} = a^3 - 2abc \end{align}$$  But there is no readily apparent pattern and the computation becomes very difficult when $n$ gets large. Display Blank when Referencing Blank Cell in Excel 2010 How to select first occurrence between two patterns including them How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something SharePoint 2013 Online - Missing Follow Link Integrated circuits seem to have standard voltages of 5 V, 3.3 V, 2.5 V, 1.8 V, etc.  Who decides these voltages? Why do smaller devices require lower voltages? How do I disable Bootchart? I dont want a Bootchart to be generated on every startup. How do I disable bootchart ?  Once disabled, how do I re-enable bootchart ?  I don't mean uninstall. "I who has" vs "I who have." Possible Duplicate:          Am aware it is correct to say "I have written..." and "I am a doctor who has writen..."  I have, however, severally come across the sentence "I, Tertius, who have written...". Shouldn't it be "I, Tertius, who has written..."? Can Mage Hand be used in combat to grab something from an opponent?  For example, one could grab a bow out of their hands or the arrows from a quiver, or try to push them out of a tree, etc.  Mage Hand can carry up to 10 pounds, but what is its ability to interact in situations like these? I tried this way, I only need to know if this is correct or if there are better ways to solve this:  $2^{1000}$ does not have a factor of $5$ obviously therefore we can assume  $$ 10^{m} &lt; 2^{1000} &lt; 10^{m+1}$$ for some  $m$   Assume $ k = 2^{1000}$, then take log on both sides $\log k = 1000  \log 2 \approx 301.02999 &gt; 301$  Therefore $2^{1000}$ has $302$ digits. Recursive cast from SimpleXMLObject to Array I was reviewing  for , and noticed that nobody voted to close it, even though the page reports a duplicate.      Why does this happen?