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This is another simple reduction i'm not quite getting: (Problem posted below in a moment).
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Simplifying from \[mv _{_{i}}=(\frac{ 1 }{ 2 }m)(2v _{i})\cos \theta _{1}+(\frac{ 1 }{ 2 }m)(2v _{i})\cos \theta _{2}\]to\[\cos \theta _{1} = \frac{ 1 }{ 2 }\]It seems so damn simple already, how do they somehow remove the other term with cosine? By the way, the two thetas are equal and opposite in value.
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