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Unart
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Homework Statement
What is the velocity vertor of a particle traveling to the right along the hyperbola y=x-1 which constant 5 cm/s when the particles location is (2, ##\frac{1}{2}##)?
Homework Equations
The Length of path forumula.
$$ s\,=\int_a^b ||r'(t)||\,dt $$
Please don't make me write you the magnitude formula, LaTeX is a pain, it's going to take me a while to type out everything.
3. Relevant example
This is an example finding the Arc Length which is need it order to find the unit velocity vector of the Arc Length. There were a few steps I didn't understand well. I'll point them out to you. I have a feeling is probably some Calculus 2 or 1 I'm forgetting.
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Find the arc length Parametrization of r(t) = <t2,t3>
assume t>0
r'(t) = <2t,3t2>
||r'(t)|| =(4t2+9t4)1/2 ##\Rightarrow## t (4+9t2)1/2
So now we integrate...
S(t) = ##\int_0^t t(4+9t^2)^{1/2}\,dt## u=t2; du=2t dt ##\Rightarrow\frac{du}{2} = t\,dt##
This is where he completely looses me. How did t become t2? Where does the 1/9 come from, is this like as if he did u=9t2 instead?
S(t) = ##\frac{1}{2}\int_0^{t^2} (4+9u)^{1/2}\,du##
S(t) = ##\frac{1}{2}\cdot\frac{2}{3}(4+9u)\cdot\frac{1}{9}\int_0^{t^2}##
S(t) = ##\frac{1}{27}[(4+9(t)^2)^{3/2}-8]##
Invert
27s = (4+9t2)3/2-8 ##\Rightarrow## 9t^2=(27s+8)2/3-4
t2=(s+ ##\frac{8}{27}##)-##\frac{4}{9}##
Lastly, how did he get these coordinates?
δ(s)= ##\left( (s+\frac{8}{27})^{2/3}-\frac{4}{9},[(s+\frac{8}{27})^{2/3}-\frac{4}{9}]^{3/2}\right)##
4. The attempt at a solution
This problem has little to do with this problem other than I have to do the first step in order to find the velocity vector at said point on the curve when the velocity is a constant 5cm/s.
if X=t then y=t-1 ##\Rightarrow\hspace {30mm} R(t)=<t,t^{-1}>##
R'(t) = <1,-t-2>##\Rightarrow## ||R'(t)||=(1+##\frac{1}{t^4}##)1/2
##\int_0^t (1+\frac{1}{t^4})^{1/2})\Rightarrow u=t^{-4}\, and \,du= -4t^{-5} dt\, or\, \frac{du}{-4t^-5}=dt##
which leads to
##\frac {-4}{t^{5}}\int_0^t (1+u)du##
5. Summary
So, did I do something wrong? Where do I go from here? And, could someone help explain what the teacher did?
Please and thankyou! :-)