How can I get an equation of trajectory of a particle mathematically for circular motion? How can I get an equation of trajectory of any particle moving in a circular motion? I mean I want to get an equation of a circle by using the conditions given for a circular motion. We know that if a particle enters in a force field $\vec F$ with velocity $\vec v$, where force $\vec F$ is perpendicular to $\vec v$, and if the force remains perpendicular to $\vec v$ always, then the trajectory of that particle becomes a circle. We can understand it physically. $\vec F$ is always perpendicular on $\vec v$ and hence on the displacement at every position of the particle. We know the force is always directed to a fixed point from which the force is created. At any point on the trajectory, the direction of displacement is given by the tangent on that point. so we can assume the force as the radius of a circle, as it is always perpendicular to the tangent on the trajectory, which is a property of a circle. So the trajectory will be a circle. But how can I get it mathematically? Can I get an equation of a circle for the trajectory of the particle by using the conditions(like $\vec F$ is always perpendicular to $\vec v$ and hence on displacement or something like that)?
What will be the trajectory of the given motion If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?   I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.  Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that.
How can I access a JavaScript object which has spaces in the object's key? How and when is old-releases.ubuntu.com updated? A customer has a 6-screen setup for each workstation and would like various pages of a web app to be opened automatically in each of the screens on startup or invoking a shortcut. (The target web browser is likely to be Chrome).  I wondered whether it would be possible to use Javascript window.open to do this but one possible snag is the customer may wish to have tabbed browsing with popup blocker turned on. (OK, these could be overridden but such config is currently out of our control...)  So the question is: Could a script (e.g. batch file) be written to open multiple instances of the browser at specified coordinates within the 6 screens (each pointing to different URLs)?  [This is basically the same question as  but for Windows instead of Ubuntu.] How to do integer & float calculations, in bash or other languages/frameworks? Is "funnest" a word? Is there an off/disable control anywhere? I realize that the INFO button toggles it off, but it comes right back on as soon as you tap the shutter, and that it a problem in low light situations where the LCD keeps blasting me in the eye. Proving this can be done as follows: consider a finite group G and elements $g_i \in G$ for some integer $i$. Now consider $\langle g_i \rangle = \{g_i^n: n\geq 0\}$, a generator. It can be proved that $\langle g_i \rangle \leq G$ and that the order of $g_i$ is equal to the order of $\langle g_i \rangle$, so $|\langle g_i \rangle| \leq |G|$.  We can now use Lagrange's theorem which states that if $H \leq G$ then $|H|$ divides $|G|$ and we're done. But...  Is there a simpler way to prove this fact? I downloaded Heroku from Heroku toolbelt using  wget -qO- https://toolbelt.heroku.com/install-ubuntu.sh | sh   in the terminal, which they recommended on their site.   My Ubuntu (14.04.1) didn't recognize the -q0-, so I took it out.  It then seemed to download fine - I see the  in the directory I was in when i downloaded it.   Now, however, it still doesn't recognize heroku as a command.  It also won't show me where it is with the 'whereis' command.  But when I look in the current directory with ls -a, it shows up.  It seems to be there.   How do I install it? I am visiting Vietnam in November 2015, I will entering the country on the 10th and my flight home is 14 days later, I will be leaving Vietnam and travelling to Cambodia in between the 2 weeks then travel back into Vietnam to get my flight home on the 24th November, do I need a visa? Can't get hold of anyone at Embassy and not clear about entering twice on the website. Book about a teenager programmed to be an assassin and trying to get back at the people responsible I read this book a few years back (around 5?). It was part of a multibook series but I have only read one book.  It's about this teenager who I think had been programmed to be an assassin but then somehow retained his conscience and was out to get the people responsible for that.  I briefly remember a girl, and this one part where he realizes he can ride a horse and then there's a chase too. I think the guy's name was Jim or Jimmy, but not quite. I'm also quite sure he was either part computer, or had a chip or something in his brain which had been programmed to give him a host of mental and physical abilities which he retained and used as well as the series progressed. Rainbow Six Siege Insane Load times? Computer or Internet? I have a high-end computer with decent ethernet yet whenever I join a game in Siege everyone finishes loading and it takes me 1-5 minutes. Is this because of my internet or computer?   Specs:  CPU - i7-7700k GPU - GTX 1080 16GG RAM 5900 RPM 4TB Hard drive Here is my problem: I've got a dialog with some parameters that the user can change (via a spinner for example). Each time one of these parameters is changed, I launch a thread to update a 3D view according to the new parameter value.  If the user changes another value (or the same value again by clicking many times on the spinner arrow) while the first thread is working, I would like to abort the first thread (and the update of the 3D view) and launch a new one with the latest parameter value.  How can I do something like that?  PS: There is no loop in the run() method of my thread, so checking for a flag is not an option: the thread updating the 3D view basically only calls a single method that is very long to execute. I can't add any flag in this method asking to abort either as I do not have access to its code. Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\mathfrak{m})^\mathfrak{m} = 2^{\mathfrak{m} \cdot \mathfrak{m}} = 2^\mathfrak{m} $$ The last step relies much on the axiom of choice, because the fact $\mathfrak{m}\cdot\mathfrak{m} = \mathfrak{m}$ does. Moreover,  states that in $\sf{ZF}$ this fact is equivalent to $\sf{AC}$.  Note, by a cardinal I mean an equipollence class of an arbitrary set, not necessarily admitting a well-ordering.  That has motivated my question: what can we say about relationship of the initial equality and the axiom of choice in pure $\sf{ZF}$?    To avoid question about definition of cardinals without $\sf{AC}$, we may consider the following statement instead:     For any infinite set $X$ exists a bijection $\; F \, \colon X^X \rightarrow 2^X$   where $X^X$ is the set of all functions from $X$ to $X$, $\; 2^X = \mathcal P(X)$ is the power set, and an infinite set is such that cannot be bijectively mapped onto a finite ordinal. I'm using tex4ht to produce XHTML files from a LaTeX file. The XHTML file I obtain uses images for some equations which tex4ht could not convert directly to XHTML.   Problem: images of equations have very bad quality.   What I want: Better quality images.  More info on attempts &amp; system:  I'm using MikTeX 2.9 under Windows Vista. I go to the command line and type htlatex document.tex "xhtml" to get the resulting XHMTL file.   After doing some search on the web, I tried to change the file tex4ht.env as mentioned in  (changing -density 110x100 to -density 220x220 -geometry 50%) in all locations I could find in my system (within the MikTeX path and also in the AppData windows directory) but the results are the same. I obtain during the compilation of hlatex the following line (among others):     System call: mgs -sDEVICE=pngalpha -sOutputFile=Untitled11x.png   -r110x110 -dEPSC rop -dBackgroundColor=16#ffffff -dTextAlphaBits=2 -dGraphicsAlphaBits=2 -q -dbat ch -dNOPAUSE zzUntitled1.ps -c quit   In particular, I would guess that the -r110x110 means resolution was still set to 110x110, despite my changes to the file tex4ht.env.  Any suggestions on how to improve the quality of equations images? How does extra damage work for critical hits? I'm a bit confused on how extra damage works for critical hits. Let's take the following example: A thief with sneak attack rolls a crit. I'm under the impression he rolls double his normal attack damage dice AND double his Sneak Attack damage dice. Is this accurate? Another example, I finished DM-ing my first campaign recently. The Giant Spider enemy monster has an melee attack that causes a constitution saving throw. If the player fails, the enemy rolls an additional 2D8 of damage. In this situation, if there was a critical hit, and the player failed their saving throw, would I also roll double dice on the additional damage? Or is that considered a separate attack? Free UML drawing library for .NET Failed to download repository information due to missing CDROM When updating I got this error message:  W:Failed to fetch cdrom://Ubuntu 11.10 _Oneiric Ocelot_ - Release amd64 (20111012)/dists/oneiric/main/binary-i386/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs W:Failed to fetch http://ppa.launchpad.net/synce/ubuntu/dists/your_ubuntu_version/main/source/Sources  404  Not Found   What to do? Selenium/PhantomJS raises error tcolorbox manual tip box, how to clone? Determine if wire in switch box is safe to re-purpose? I need to pull a neutral wire to my light switch for a smart switch.  It's presently switching an outlet that I don't need switched, and the other side of the outlet already has unswitched power.  My plan is to use a smart switch, which requires neutral, to control a different device wirelessly.    I can't feasibly run a new wire, because the wires aren't run through conduit.  Instead, my plan is to repurpose the switched wire to get neutral from the outlet.    The switch, which controls an outlet has two wires, one black and one red.  The switched outlet has a white neutral wire, plus the red and black wires that I assume run directly to the switch.  I've completely disconnected the red and black wires from the outlet and the switch.  The black wire is measuring 120V AC when measured against ground using the box as ground, both at the outlet and at the switch. Using my DMM, the red wire is measuring 24V AC with respect to the box/ground at both sides.  I assume this is a  caused by induction from the charged black wire over to the floating red wire.    Our house is ~100 years old and this particular switch seems to be run using armored cable (note the paper wrap around each wire.)  There are literally only two wires in the switch box; no wire nuts.    If relevant, we do have some old knob and tube wiring active, but I'm pretty confident there's none on this circuit.  Is there some way I can determine with enough confidence that the red wire isn't connected elsewhere such that I can know it's safe to connect the red wire at the outlet to netural?