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See here, when I put v = 0 on the bottom,
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because the top had a v, not v/c,
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something was left over
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that looked like something familiar, namely momentum.
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Here, if I take v = 0 too quickly,
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what's left over is nothing reminiscent of anything in Newtonian mechanics.
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So, you want to go in
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what's called the next order in v/c.
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You don't want to totally ignore it.
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You want to keep the first non-zero term,
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so we write it
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I've just written it with