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it's proportional to sin(t)
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and a t = 0, that vanishes, that's correct;
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you pulled it and you let it go.
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So, the instant you released it, it had no velocity.
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And it satisfies Newton's Laws, and that's your answer.
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I want to do this simple example in totality
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because this is the paradigm.
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This is the example after which everything else is modeled.
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Pardon me?
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You mean like t = 0, if the answer was sin(t),
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where sin (t) vanishes at t = 0.
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And cos(t), it doesn't vanish, has a robust value of 1.