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The information to the question is, "A $59.0~kg$ boy and his $38.0~kg$ sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with a velocity $3.00~m/s$ toward the west. Ignore friction."

The specific question I am working on is, "How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system?"

Well, I said that $E_{mech,~i}=0$, because no one has kinetic energy (no one is moving), and neither of them have the potential to move, that is, until the girl commences with the push; but, $E_{mech,~f}=PE=KE$, when she uses her potential energy to apply a force over a distance, on her brother, and by doing so she changes the energy of the system, by changing the speed of both of them, meaning the potential energy will convert to kinetic energy.

I've tried to plug every piece of data given to me, but I still couldn't find how much $PE $ was converted to mechanical energy. How do I find it? Is my analysis correct; is there anyway to improve it?

Edit:

I have another question: why can't internal forces of a system cause momentum to not be conserved?

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You have this equation:

$m_bv_b+m_gv_g=0$ (conservation of momentum)

and $m_b,v_b,m_g$ are given. You can easily solve for $v_g$. Once you have this, $|\Delta PE|=\Delta KE_f=\frac12m_bc_b^2+\frac12m_gv_g^2$

I have another question: why can't internal forces of a system cause momentum to not be conserved?

Who said that they can't? Momentum is always conserved when there are no external forces on the system.

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@Manshearth No, I was asking why internal forces don't affect the total momentum of a system. –  Mack Nov 17 '12 at 17:57
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