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A sphere of mass $m_1$ moving with speed $u_1$ collides directly with a similar sphere of mass $m_2$ moving with speed $u_2$ in the same direction, with $u_1>u_2$. The coefficient of restitution between the two spheres is $e$. Show that the loss of kinetic energy $E$ due to the collision satisfies the equation
$2(m_{1}+m_{2})E=m_{1}m_{2}(1-e^{2})(u_{1}-u_{2})^2$

And here's the link for the paper with the projectile q. (q11) :
http://www.mathshelper.co.uk/STEP%20II%202006.pdf

Aye - this is always a good technique, what I tend to do is use the explicit sub $u=xy$ and $v = x + y$ (or more generally, $u = f(x,y)$ and $v = g(x,y)$), it is usually the intended solution to spot that the pattern in both equations is the same, so to speak and to exploit this fact. Anywho, this gives us a linear system in $u,v$ which is easier to solve.

I like making the explicit sub because it's easier to just power through the algebra, it saves some time re:writing xy and x+y over and over again, although admittedly not much here but it certainly would in other problems where either of $f(x,y)$ or $g(x,y)$ is nasty and I find that it helps avoiding silly mistakes as well.

Aye - this is always a good technique, what I tend to do is use the explicit sub $u=xy$ and $v = x + y$ (or more generally, $u = f(x,y)$ and $v = g(x,y)$), it is usually the intended solution to spot that the pattern in both equations is the same, so to speak and to exploit this fact. Anywho, this gives us a linear system in $u,v$ which is easier to solve.

I like making the explicit sub because it's easier to just power through the algebra, it saves some time re:writing xy and x+y over and over again, although admittedly not much here but it certainly would in other problems where either of $f(x,y)$ or $g(x,y)$ is nasty and I find that it helps avoiding silly mistakes as well.

Subtracting expressions during Induction:
Consider the recurrence $H_{n+1}(x)+2xH_{n}(x)+1=0$ where $H_{n}(x)$ is some predefined function. If we wanted to prove $H_{n+2}(x)+(H_{n}(x))^{2}=f(x)$ for a given f(x) for example, it is often easier to consider $H_{n+2}(x)+(H_{n}(x))^{2}-(H_{n+1}(x)+(H_{n-1}(x))^{2}$ for example in the inductive step (as here we just need to show the resulting terms=0). (This is just a random example so may not give anything sensible in this case!)

Do you mean doing a concrete example of this? I'm not sure where the multiple squares are coming from here.

Yeah sure! Let me refer to my good friends the chebyshev polynomials defined by $T_{0}(x )=1, T_{1}(x)=x , T_{n+1}(x)=2xT_{n}-T_{n-1}$.Then we can prove $(T_{n}(x))^2-T_{n+1}(x)T_{n-1}(x)=(T_{1}(x))^2-T_{2}(x)T_{0}(x)$ if you try the method above it comes out a little neater than other methods

You're given the expression and you want to prove it, right? You're not meant to conjure it out of thin air?

Hey Insight, did I remember correctly that you are doing m4/m5?

True that.

Please would you be able to look over a strange solution for a question in a markscheme for me? I really can't see what they did, but it only took them about 1 line of working and drawing a couple of lines. Its q2 on the june 05 paper (https://013960f3b9a5a7ae938f512b8c85e3fe8578e56e.googledrive.com/host/0B1ZiqBksUHNYVF9pRzgzbXd2YkE/June%202005%20QP%20-%20M4%20Edexcel.pdf) if you wouldnt mind looking some time itd be appreciated

This was quite an easy question. What happened?

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I got the correct answer but was unsure of why we could construct that vector triangle like we did. So all I needed is that qVp is perpendicular to vq, and using similar triangles we can form that triangle drawn inside the circle? (lol sorry for my ignorance to these questions, ive never seen any of this written down anywhere so have to try and derive it first time when doing papers). But yeah Ive covered m5 and Im happy with that, seems easier than m4 in a way as its more pure maths inclined. The only questions I seem to mess up are the 'find the direction of velocity to have distance between these two objects as small as possible type questions". Thanks for the help!

Also, have you finished Mechanics 5? Or are you currently practicing Mechanics 4 before starting M5?

I am kind of fine with Relative Motion right now, but need to practice questions on Stability and maybe Damped Harmonic Motion, I still struggle to understand how extensions work in M3 SHM sometimes but there is enough time to grind papers before those exams.

Don't you have the M4 textbook?







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