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fluidistic
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Homework Statement
I've found in a physics book that when u <<L, we have [itex]\sqrt {u^2+L^2} -L = \frac{u^2}{2L}[/itex]. I don't understand why.
Homework Equations
Not sure.
The Attempt at a Solution
I've calculated the Taylor expansion of order 2 of [itex]\sqrt {u^2+L^2} -L[/itex] but I couldn't show the approximation. I get 0, which is "non sense".
If [itex]f(u)=\sqrt {u^2+L^2} -L[/itex], then [itex]f'(u)=\frac{u}{\sqrt {u^2+L^2 }}[/itex] and [itex]f''(u)= \frac{1}{u^2+L^2} \left ( \sqrt {u^2+L^2}- \frac{u^2}{\sqrt {u^2+L^2} } \right )[/itex].
So [itex]f(u)\approx f(0)+f'(0)u+ \frac{f''(0)u^2}{2}[/itex]. All of these terms are approximately worth 0 (the last one) or simply worth 0 (the 2 first).
I'm totally stuck on this.
Edit: Nevermind, I figured it out. The 3rd term (the one that isn't worth 0) is worth precisely [itex]\frac{u^2}{2L}[/itex]. I considered that u^2 was almost worth 0 at first, because in physics when we have a differential squared we consider it as 0... And here u is considered very small so I got confused, but I finally "understand" the solution now.
Thanks for reading tough. Hope that can help someone else in future.
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