Deriving speed equations | Elastic collision

[I_Try_Math]

Homework Statement

Derive the equations giving the final speeds for two objects that collide elastically, with the mass of the objects being $m_1$ and $m_2$ and the initial speeds being $v_{1,i}$ and $v_{2,i}=0$ (i.e., second object is initially stationary).

Relevant Equations

$p_i = p_f$

So far I've got:

$p_{1,i} + p_{2,i} = p_{1,f} + p_{2,f}$

$p_{1,i} + 0 = p_{1,f} + p_{2,f}$
$m_1v_{1,i} = m_1v_{1,f} + m_2v_{2,f}$

According to the textbook, the final speeds should be written in terms of $v_{1,i}, m_1, and m_2$. It looks like I need another way to relate everything with an equation to solve for the final speeds but I'm drawing a blank. It doesn't seem like I can use energy considerations? Any hints are appreciated.

 

[haruspex]

. It doesn't seem like I can use energy considerations?

Why not?
" two objects that collide elastically"