Two particles of same mass in a 2D frame collide with known initial (i) velocities. I would like to know the final (f) velocities of them after the collision.  As in any other collision, momentum is conserved after the collision. Writing in components:  $$ v_{x,1}^{i} +  v_{x,2}^{i} = v_{x,1}^{f} +  v_{x,2}^{f} $$ $$ v_{y,1}^{i} +  v_{y,2}^{i} = v_{y,1}^{f} +  v_{y,2}^{f} $$  The total energy (not the mechanical) is also conserved. K accounts for the thermal energy. $$ v_{x,1}^{2,i} +  v_{y,1}^{2,i} +  v_{x,2}^{2,i} +  v_{y,2}^{2,i}= v_{x,1}^{2,f} +  v_{y,1}^{2,f} +  v_{x,2}^{2,f} +  v_{y,2}^{2,f} + 2 \cdot K/m $$  I obtain with this 3 equations for 4 unknown quantities, the x and y components of the velocities of particle 1 and particle 2.  How could this be solved?. What information should I add? I can only think of modelling the collision with a potential of some kind.
I am trying to calculate the final velocities of two equal mass 2-dimensional circles after an elastic collision. I have tried to figure it out using formulas I know from high school physics, but nothing seems to work. My known variables are:   initial x and y velocities for each circle x and y distances between the centres of the circles at the time of the collision   The variables I would like to solve for are:   final x and y velocities for each circle   Here are diagrams showing all the variables:
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So I wanted to create a &quot;big list&quot; of equivalent definitions of convex functions, I will start with some equivalent definitions and post my proofs in my answer below. Feel free to add any other equivalent definitions. This question can then serve as an abstract duplicate. Claim. Let $n\in\mathbb N$ and $f\in\mathcal C^1(\mathbb R^n)$. Then the following are equivalent:  $f$ is convex, i.e. $f(\lambda x_1 + (1-\lambda x_2))\le\lambda f(x_1)+(1-\lambda)f(x_2)$ for all $\lambda\in[0,1]$ and $x_1,x_2\in\mathbb R^n$. For all $x,y\in\mathbb R^n$, $\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq0$. For every $x,y\in\mathbb R^n$, $f(y)\geq f(x)+\langle y-x,\nabla f(x)\rangle$. How to derive addition of velocities without the Lorentz transformation? Lorentz contraction and time dilatation can be deduced without Lorentz transformation. Can you deduce also the theorem of    $$w~=~\dfrac{u+v}{1+uv/c^2}$$   without Lorentz transformation? Using just the constancy of light speed. Create a text file using user input from a batch file Is it possible to write a batch file that will create a popup window for the user to input some specific information, and place that information into a text file? The text file should already have some text in it but I just want a couple lines to have input from the user. This is what the text file should look like:  PARTSN="USER INPUT1" PART=blahblah WORKSTATION=blahblah USERID="USER INPUT2" SIZE=blahblah   Where it says "USER INPUT1,2" is where I want the information from the user placed.  Any help would be great! Number theory proof mod p For a given prime $p$ and integer $k$,  $k∈&lt;1, p-1&gt;$ $S(k)$ is defined as a sum of remainders of $k, k^2, k^3, ..., k^{p-1}$ when divided by $p$. Prove that $S(k)=\frac{1}{2}p(p-1)$ is true for an odd number of different k. I managed to prove that $S(p-1)=\frac{1}{2}p(p-1)$ but can't see how to do it further. 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What happens when your max HP is 0 because you leveled up with a negative Constitution modifier? Are you dead, unconscious, or what, and where do the rules say this? (The  assumes death, but that isn't established as fact anywhere yet.) Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$   then p can't be written as a sum of two squares.  Thanks! Obviously, this suggested edit should never have been approved. Because it's very, very minor.  Is it acceptable for me to "rollback" this edit? Does the suggested editor lose the +2 rep when I roll it back? Should I comment on one of his questions, explaining not to make too minor edits like this?  Is there anything I should do to notify the reviewers that approved  too minor suggested edit?