Not a factor of ten, a multiple of ten. ...The way that kinetic energy and momentum work, you can conserve one or the other but not both. ...
I fail to see any difference between "a factor of" and "a multiple of" ten, but I will not argue symantics with you.
Instead, as you continue to stubbornly display your stupidity, I will now explicity show where the second of your two errors is. Because you have a false assumption, an "error of analysis," you computed 10, for the energy gain ("multiple") instead of 2.5, but that is still impossible and due to your lack of understanding of the system. (Which is your first, and more serious error. - I may explain that to you later.)
OK, Below I do the analysis of your FALSELY UNDERSTOOD system correctly for you. I have waited for you to correct your mistake, but doing this analysis correctly seems to be beyond your capabilities.
Like you, I will assume an idea see-saw, i.e. it has no mass and yet is completely rigid and does not absorb any energy.* My notation is easy to remember: All small letters refer to the small mass, m, even though its speed after contact, s, is ten times greater than the big mass’s speed, S, due to fact mass m is sitting initially on the see-saw's lever arm that is 10 times longer. I.e.
s =10S and also by your assumption,
M =10m gives the relationship between the two masses. Before contact the speed of m is zero, i.e. v = 0, and speed of the big mass is called V.
For notational convenience, I measure energy in “half Joule” units. (This avoids the usual 0.5 factor.)
Thus, T1, the Total energy input to the system is MV^2; and after M's contact with the see-saw:
The total energy, T2 = MS^2 + ms^2 = MS^2 + (M/10)(10S)^2 = 10MS^2.
You think that the energy gain, G = (T2 / T1) = 10(S / V)^2 is 10, because you falsely assume that S = V. This S = V error is but ONE of your errors.
You cannot even analyze the wrong system correctly! S < V because the small mass has acquired kinetic energy. Conservation of energy implies that the velocity of the big mass must become less. But in post 89 you foolishly state:
... the law of conservation of energy, is false. Conservation of momentum is true....
They are both true.
I will now do your false system's analysis correctly; i. e. without your false S = V assumption, but keep in mind the results are still NONSENSE, as you do not understand the system. That is your major error, but I may (or may not) explain it to you later. For now, I only provide a hint in the footnote. I.e.
I am giving you time to correct your major conceptual error by yourself.
To find the relationship between S and V, you should have applied “conservation of momentum” correctly instead of assuming S = V (but still will get the wrong result due to your other, more important, conceptual error) .
The pre-contact total momentum,
M1 = MV. The after contact the total momentum, M2, is:
M2 = MS + ms = MS +10mS = MS +10(M/10)S = MS + MS so:
M2 = 2MS.
{I show each step as I realize your abilities are limited.}
Now you apply (incorrectly) “conservation of momentum” i.e. M1 = M2 or using the two bold equations just above, MV = 2MS and,
then dividing by 2M, conclude S = V/2, which is very different from your false assumption.
Then from the above, G = 10(S / V)^2 = 10(1/2)^2 = 10/4 = 2.5, not 10.
I.e. the
impossible energy gain is not your 10, but a still NON-SENSICAL, 2.5.
-------------
*This implies an infinitely large force at the moment of contact, but as F = ma was not used in the analysis, that did not cause mathematical problems. This does provide a hint for understanding the problem correctly. (I.e. is a hint for seeing your major error.)