There is a formula for the conservation of momentum: how can p = √ 2mek be deduced, and under what conditions can it be used?

There is a formula for the conservation of momentum: how can p = √ 2mek be deduced, and under what conditions can it be used?

Ek=1/2mv²,p=mv
EK = 1 / 2m & sup2; V & sup2 / M = 1 / 2p & sup2 / m, so p = √ 2mek

What is the difference between kinetic energy conservation, momentum conservation, mechanical energy work, gravity work and kinetic energy work?
It seems to be, but I want to ask what is the change of mechanical energy? The rest is as follows

1. The change of kinetic energy depends on the work done by external force (kinetic energy theorem). If it is to be conserved, the work done by external force is 0
2. The condition of conservation of mechanical energy is that no work is done by other forces except gravity (elastic force) or the total work done is 0
3. The change of momentum depends on impulse, and the conservation condition is that the combined impulse is 0, that is, the combined external force is 0

Simultaneous derivation of conservation of physical momentum and conservation of mechanical energy~
Equation 1: m1v1 + m2v2 = m1v3 + m2v4
Equation 2: m1v1 ^ 2 + m2v2 ^ 2 = m1v3 ^ 2 + m2v4 ^ 2
Results: the two formulas were solved simultaneously
After simplifying the two formulas, I get formula 3: V1 + V3 = V2 + v4
Find and then bring formula 3 into Formula 1 to get the result! (both M1 M2 V1 is used to express V3 V4)
The answer is V3 = (m1-m2) / (M1 + m2) * V1 to process!!!

V3 = [(m1-m2) V1 + 2m2v2] / (M1 + m2) when V2 = 0, V3 = (m1-m2) V1 / (M1 + m2)
V4 = [(m2-m1) V2 + 2m1v1] / (M1 + m2) when V2 = 0, V4 = 2m1v1 / (M1 + m2)