- #1
titansarus
- 62
- 0
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.