if two masses Mass_0 with Velocity_0 and Mass_1 with Velocity_1 collide directly in opposite directions it is said that momentum is conserved so that (Mass_0)(Velocity_0) + (Mass_1)(Velocity_1) = (Mass_0 + Mass_1)(Velocity_2).

Because Velocity is a vector quantity and can be negative Velocity_2 < abs(Velocity_0+Velocity_1) therefore kinetic energy which increases with Velocity^2 of the system will be less and that difference is said to be lost as radiated heat or internal vibrations or something more exotic. Therefore total energy of the system remains unchanged because all those vibrations and heats are still part of that sum.

Now say this did not happen in a vacuum. Mass_0 was moving away from some enormous Star_0 and Mass_1 was moving away from a very tiny Star_1. Now assume KE_1 > KE_0. The sum of mass is moving toward a larger mass Star_0 and the Potential Energy is unrelated to velocity. Therefore the Potential Energy of the system has increased while the Total Kinetic Energy+ Potential energy from the collision remained constant. The system Total_Energy = KE+PE.

Therefore Total_Energy_1/ Total_Energy_0 is over unity.

If You want to see this is true just plug in some simple numbers. I used Mass_0=1kg Velocity_0=-1 m/s Mass_1=2 kg Velocity_1=2 m/s Star_0=10^30kg Star_1=10^10kg. and the equations from Newton as taught in schools. m_1M_2 G / r^2 = PE and 1/2 M V^2 = KE and Momentum = mv.

I am not a criminal stop trying to treat me like one; to preserve the conservation of energy this problem can be solved by multiplying KE = 1/2 M (or I) V^2 by a unit vector to preserve direction despite squaring velocity.

It seems simple but noone has ever thought to do it as far as I know so I get to name this Anonymous Theorem.