Two solutions for a collision which satisfy both conservation of momentum and energy. Which is correct? For the following problem, there are two sets of solution which appear to be equally valid:
v_car = 15.7 m/s, v_truck = 12.9 m/s and v_car = 12.3 m/s, v_truck = 15.2 m/s. How can I know which set of solutions is correct in such cases?
"A 1000-kg car traveling with an x component of velocity of +20 m/s collides head-on with a 1500-kg light truck traveling with an x component of velocity of +10 m/s.If 10% of the system's kinetic energy is converted to internal energy during the collision, what are the magnitudes of the final speeds of the car and truck?"
 A: To approach a more general form of this problem, let's skip the first step of your problem and just suppose you have already calculated:
Total momentum
$P' = m_1 v'_1 + m_2 v'_2$
Total energy
$E' = (1/2) m_1 v_1^{'2} + (1/2) m_2 v_2^{'2}$
Let $f = m_1/m_2$, and let's do our work in the center of mass frame where P=0. And instead of conserving energy E, I'll use a convenient constant C which is proportional to the center-of-mass energy, to save some factors of 2 and $m_2$. For ease, the velocities $v_1$ and $v_2$ represent center-of-mass velocities.
Momentum:
$f v_1 + v_2 = 0$, or $v_2 = -f v_1$
Energy:
$A = f v_1^2 + v_2^2 = f v_1^2 + f^2 v_1^2$
Now $v_1 = \pm \sqrt{\frac{A}{f+f^{2}}}$.
In the center-of-mass frame, it's obvious which root to choose based on the principle that @knzhou described in his comment, which is a little more general than it sounds! You choose the opposite sign of the initial velocity in the center-of-mass frame.
And then of course to solve your actual problem, you add back the center of mass velocity.
