What if Only Momentum is Conserved?

Both energy and momentum always conserved

Two weights, about to fall from their highest position

This is just a "what if" post. Every physical process known (at least to me) always conserves both energy and momentum. But let's consider the two weights shown, on arms of negligible weight pivoted to the fixed axle, falling down together through 180 degrees to their lowest position. Each weight will reach a speed given by:—

      v² = u² + 2gh

where v = final velocity
            u = initial velocity = 0
            g = gravitational acceleration = 9.80665 m/s²
            h = total height of fall

If we set h = 1 meter, then we get

      v² = 0² + 2 × 9.80665 × 1

giving v = 4.4287 m/s for each weight when it has fallen down.

If each weight has a mass of 4kg, then, for the system as a whole:—

Total kinetic energy (½mv²) = 2 × ½ × 4 × 4.4287² = 78.45 J.
Total momentum (mv) = 2 × 4 × 4.4287 = 35.4296 kg-m/s.

An immediate transfer of momentum 

Let's also suppose that when it has fallen to its lowest position, one of the weights can immediately give up all of its momentum to the other weight. The first weight will obviously stop instantly, and from conservation of momentum, the still-moving weight must now have a velocity equal to the total momentum divided by its mass, i.e. 35.4296 ÷ 4 = 8.8574 m/s. So now the moving weight's kinetic energy is ½ × 4 × 8.8574² = 156.9 J, i.e. it now has twice the total energy that was gained when both the weights fell. That would certainly be an energy increase worth having!

Could it be done?

Could any process be found that would permit a weight to transfer all its momentum to another weight in such a way as to gain energy like this? I must admit I can't think of any, although I can understand why some people like to look into this idea.