Maximizing Transferred Energy between n bodies in linear collisions

[titansarus]

Homework Statement

My problem has two parts.
1) We have two point masses $m,M$. and there is another mass $m_1$ between them.They are all aligned in a line. Mass $M$ is moving with speed $u_1$ toward $m_1$ and after collision and all other masses are not moving. we want to find $m_1$ such that the kinetic energy of $m$ get maximum.
2) Now think that we have n masses $m_1 ,m_2 ,... m_n$ between $m , M$. find $m_1 , m_2 , ... , m_n$ such that the kinetic energy of $m$ get maximum. (elasticity coefficients are $e_1$ and $e_2$ ,... but they are not actually important and you can ignore them i.e. $e=1$)

Homework Equations

In collision between two masses of mass $m_1$ and $m_2$ with elasticity coefficient $e$ moving with speed $u_1$ and $u_2$, speed of $v_1$ after collision is:
$v_1 = \frac{(m_1-m_2 e)}{m_1+m_2} u_1 + \frac{m_2 e}{m_1 + m_2} u_2$ (Eq.1)

The Attempt at a Solution

[/B]
The main goal is to maximize $v$ of $m$. For part 1 if we write the (Eq.1) two times with $u_1 = 0$ for $M,m_1$ and $m_1,m$ we get:

$v_m = 4 ~e~e'~M u_1 (\frac{m_1}{(m+m_1)(M+m_1)})$. If we calculate $\frac{d}{d~m_1} (v_m) =0$ we get $m_1 =\sqrt {mM}$.

For the second part, I don't know how to formally write a proof but I think the answer will be $m_i = \sqrt{m_{i-1}m_{i+1}}$. which means that every mass must be the geometric mean of its left and right mass and for $m_1$ we get $m_1 = \sqrt{M m_2}$ and for $m_n$ we get $m_n = \sqrt{m m_{n-1}}$. I think it is somehow obvious that we must maximize the $v$ in each collision but I can't mathematically prove that. So I want to know how to write a formal proof for this question.

 

[haruspex]

So I want to know how to write a formal proof for this question.

Suppose you think you have somehow managed to tune all the intermediate masses so as to get the biggest final velocity you can. What if you find three adjacent masses in the sequence that do not fit the expected pattern. What can you deduce?

 

[titansarus]

Suppose you think you have somehow managed to tune all the intermediate masses so as to get the biggest final velocity you can. What if you find three adjacent masses in the sequence that do not fit the expected pattern. What can you deduce?

Maybe we can say that if this thing happens for $m_{i-1} , m_{i} , m_{i+1}$ and they do not fit the expected pattern, then if we consider a $v$ for $m_{i-1}$, then the energy (velocity) transferred to $m_{i+1}$ will not be maximum. However, I think maybe there should be a way to solve this without using "Proof by contradiction". Perhaps there is way using "Mathematical Induction" or even a direct proof like the derivative (max-min) proof for 3 mass as I said in the first post. But I cannot write it in a formal way for n-body.

 

[haruspex]

should be a way to solve this without using "Proof by contradiction".

It's a perfectly honourable style of proof.

direct proof like the derivative (max-min) proof for 3 mass

That could get messy. You have n independent variables, so n differentiations to produce n simultaneous equations.
You could get lucky.