You might explain what you are trying to say.

I assume you are saying that the energy of the .01 kg mass dropped 1 meter (v = 4.429 m/sec) is equal the energy of the 1.110 kg Atwood's moving .41666 m/sec: which is quite correct.  ½ m v²

I am not proposing that you give the energy of the 1.110 kg to the .01 kg I am proposing that you give the momentum of the 1.110 kg to the .01 kg

The 1.110 kg has .4625 units of momentum; if you give all of that momentum to the .01 kg it will be moving .01 kg * v = .4625 kg m/sec.   v = .4625 kg m/sec / .01 kg = 46.25 m/sec.

At 46.25 m/sec it will rise 109 meters. It was dropped 1 m.

Cylinder and spheres machine give the motion of a massive cylinder to much smaller mass.

Their momenti are not separate, but part of one system.


In 238 years you are not the first person to walk this path

v = (sqrt)[19.6(1.11-0.1)/(1.11+0.1)]
———————————————————
v = 4.04479055 m/s

Name the others that have walked this path.

To transfer momentum from a large mass to a small mass is incredibly simple.

But; name the others that have walked this path.

v = (sqrt) [19.6  .01 kg /1.110 kg= .4202 m/sec   ideal

The experimental value was very close  .41666 m/sec

The momenti are one and the same object, they are linked.
It is not two separate momenti (one for each mass),
But rather one single quantity, from both interconnected masses in motion.
If the large mass suddenly becomes disconnected (no transfer of momentum),
the smaller mass is accelerating downwards at 9.8m/s/s
its' upward momentum being instantly disconnected as well.
Cut the string


If a scenario were present such as an impact lever at the very bottom,
point of maximum velocity, such that the large mass hit and transferred
100% of its' momentum into an upward thrust, sending the smaller mass skyward.
And you were to catch it at its uppermost peak::::: even with 0 losses from wind or anything
(pay attention here!!)


The Potential Energy in the small mass:
E=m[small] * 9.8m/s/s * height = m[large]*9.8m/s/s* initial height


Do the tests yourself


This has been done, literally, millions of times