I think i’m missing something. When two object is colliding and no energy is lost why is the outcome determined by the total momentum formula? (m1v1 +m2v2 = m1v1′ + m2v2′)
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Why is it not total kinetic energy ? (¹/² m1.v1² + ¹/² m2.v2² = ¹/² m1.v1²’ + ¹/² m2.v2²’)
Why do we say momentum is conserved, instead of total energy is conserved?
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The solution to momentum being conserved is actually the solution to both equations.
m1v1 + m2v2 = m1v3 + m2v4 even if we know m1, m2, v1, v2, has infinite solutions because we only have 1 equation and 2 unknowns, so to.solve it we need 1/2(m1v1^2 + m2v2^2 ) = 1/2(m1v3^2 + m2v4^2). On its own, it also has infinite solutions, but with the momentum equation, we get exactly one solution (2 equations and 2 unknowns)
Usually when doing momentum conservation problems, the collision is either perfectly elastic (no energy is lost) resulting in both equations needing to be used. Or perfectly inelastic (maximum energy is lost) where the objects stick together m1v1 + m2v2 = (m1+m2)v3 which is just one equation and one unknown. Now that energy isn’t lost, but it’s given off as heat or sound or fragments of the objects breaking off and flying away. Now no collision is perfectly elastic or inelastic, so in the real world we would use m1v1 + m2v2 = m1v3 + m2v4 and/2(m1v1^2 + m2v2^2 ) = 1/2(m1v3^2 + m2v4^2) + H where H is the energy lost to heat in the collision. Of course, this gives us another unknown, which means there’s infinitely many solutions again. If we can predict the amount of heat that will be lost (based on the nature of the materials colliding) we can get a very good estimate of how the collision will play out.
So the answer is basically that we use both.
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