I am reading, in my free time, "Fundamentals of Physics" by Walker | Haliday | Resnick; Tenth edition.
I am reading chapter 9 section 9-7: "Elastic Collisions in one Dimension"
We start off with these equations:
$m_1v_{1_i} = m_1v_{1_f} + m_2v_{2_f}$ (linear momentum) (9-63)
$m_1v_{1_i}^2 = 1/2m_1v_{1_f}^2 + 1/2m_2v_{2_f}^2$ (kinetic energy) (9-64)
"To do so, we rewrite Eq. 9-63 as
$m_1(v_{1_i} - v_{1_f}) = m_2v_{2_f}$ (9-65)
"and Eq. 964 as
$m_1(v_{1_i} - v_{1_f})(v_{1_i} + v_{1_f}) = m_2v_{2_f}^2$ (9-66)
"After dividing Eq.9-66 by Eq.9-65 and doing some more algebra, we obtain
$v_{1_f} = \frac{m_1 - m_2}{m_1 + m_2}v_{1_i}$ (9-67)
and
$v_{2_f} = \frac{2m_1}{m_1 + m_2}v_{1_i}$ (9-68)
Now my question is, what does "dividing Eq.9-66 by Eq.9-65 and doing some more algebra" exactly mean here?
I don't know how they 'divided' those equations to get those simultaneous equations.
I've tried to set Eq.9-66 in terms of ${v_{1_f}}$ by re-arranging Eq.9-65 but I still don't end up with anything remotely what they have. Almost always I end up with too little $m_1$ or $m_2$'s.
I am not sure what math I am missing here to be able to do what they summed up as a division and "some more algebra". Could anyone point me in the right direction?
NOTE: This is not homework, I'm just reading a physics book because I want to learn physics. :) I haven't been in school since 2004.
When they say they divide 9-66 by 9-65, they're saying that you divide the left-hand side (LHS) of 9-66 by the LHS of 9-65, and the RHS of 9-66 by the RHS of 9-65:
$$ \frac{m_1\left(v_{1_i} - v_{1_f}\right)\left(v_{1_i} + v_{1_f}\right)}{m_1\left(v_{1_i} - v_{1_f}\right)} = \frac{m_2v_{2_f}^2}{m_2v_{2_f}} $$
$$ \Rightarrow v_{1_i} + v_{1_f} = v_{2_f} .$$
The "some more algebra" is just substituting $v_{1_i}+v_{1_f}$ for $v_{2_f}$ into 9-65:
$$ m_1\left(v_{1_i} - v_{1_f}\right) = m_2v_{2_f} $$ $$ \Rightarrow m_1\left(v_{1_i} - v_{1_f}\right) = m_2\left(v_{1_i} + v_{1_f}\right) $$ $$ \Rightarrow v_{1_f} = \frac{(m_1 - m_2)}{(m_1 + m_2)} v_{1_i} .$$
And substituting $v_{2_f}$ for $v_{1_i}+v_{1_f}$ into 9-66 (with an assist from 9-67 to eliminate $v_{1_f}$):
$$ m_1\left(v_{1_i}-v_{1_f}\right)\left(v_{2_f} + v_{1_f}\right) = m_2v_{2_f}^2 $$ $$ \Rightarrow m_1\left(v_{1_i}-v_{1_f}\right)v_{2_f} = m_2v_{2_f}^2 $$ $$ \Rightarrow m_1\left(v_{1_i}-\frac{m_1-m_2}{m_1+m_2}v_{1_i}\right) = m_2v_{2_f} $$ $$ \Rightarrow m_1\left(1-\frac{m_1-m_2}{m_1+m_2}\right)v_{1_i} = m_2v_{2_f} $$ $$ \Rightarrow m_1\left(\frac{m_1+m_2}{m_1+m_2}-\frac{m_1-m_2}{m_1+m_2}\right)v_{1_i} = m_2v_{2_f} $$ $$ \Rightarrow \frac{2m_1m_2}{m_1+m_2}v_{1_i} = m_2v_{2_f} $$ $$ \Rightarrow v_{2_f} = \frac{2m_1}{m_1+m_2}v_{1_i} .$$