# What is the meaning of arc length in this physical scenario?

Let's say a particle's velocity is modeled by $v(t)=\sin(t)$. Therefore, assuming the particle starts at position $0$, it's position $x$ can be modeled by $x(t)=\cos(t)$. The total distance traveled by the particle in $t$ seconds can then be modeled by $\int_0^t{|\sin(t')|}\mathrm{d}t'$. In this instance, what is the arc length of the position in physical terms? To be specific, what is the meaning of $a(g)=\int_0^g{\sqrt{1+\sin^2t}}\space \mathrm{d}t$?

• This link seems to deal with a similar question.
– Mark
Feb 27, 2016 at 5:28
• Is it meaningful in this context? The units of the terms in the square root of the arc-length integral don't match, do they?
– Neal
Feb 27, 2016 at 6:06
• In an actual physical problem, you would probably have something like $v(t) = sin(a t)$ with $a t$ being dimensionless, so that $a$ would have units of inverse time. That would take care of the subsequent dimensional issues you notice. Feb 27, 2016 at 6:54

Note that $x(t)=1-\cos t$ instead, so $x(0)=0$ and $v(t)=\sin t$.
Distance travelled $\displaystyle \, =\int_{0}^{t} |v(t)| dt =\int_{0}^{t} |\sin t| \, dt =2\left \lfloor \frac{t}{\pi} \right \rfloor +1- \cos \left( t-\left \lfloor \frac{t}{\pi} \right \rfloor \pi \right)$
Whereas $\displaystyle \int_{0}^{x} \sqrt{1+\sin^{2} x}\, dx = E(x,i)$ is the arc length of the curve $y=\cos x$