# Spatially and temporally variable velocity

I have the velocity function of an atom, $v(x,t)$, which changes with time and space. I'm looking for a general relationship for finding the location of the atom at time $t_s$. The atom start to travel from $x=0$ at $t=t_0$. I'm confused since the velocity is a function of space :-?

• Why would you to treat space $x$ and time $t$ as two independent variables for the velocity? Makes no sense to me – lemon Nov 12 '17 at 12:17
• @lemon can you elaborate more in form of an answer. I get your point but not fully. I mean I could not manage to write what I have in my mind in mathematical expressions :/ Any help is appreciated. – Dolle Nov 12 '17 at 12:20
• @lemon It seems OP is considering a continuous fluid, and asking themselves about the trajectory $\gamma(t)$ with $\gamma(0)=0$ and $\dot\gamma(t)=v(\gamma(t),t)$. If I understood it correctly, this is a rather standard excise. – AccidentalFourierTransform Nov 12 '17 at 13:06

## 1 Answer

The general relationship would be $$x(t_s) = \int_0^{t_s} v(x(t),t)\,dt$$ which follows from the definition of velocity.

Ideally you would want to eliminate the $x$ dependence in your velocity function so that $v$ is a function of time only. However, if that's not possible, then the above equation describes an implicit equation in $x$ which you would probably have to solve numerically.

• Thanks. Let me share with you a little bit more details. As you said the velocity can be written in terms of time. However, my velocity itself is coming from atomic flux ($J$) basically: $v(x,t)=\Omega J(x,t)$ where $\Omega$ is atom density. In this relationship time and space dependency is not very visible to me. I hope I explained well. Can you comment on this? I'm trying to offer a mathematical model for the phenomenon keeping the physical meaning visible in the math. – Dolle Nov 12 '17 at 12:32
• @Dolle I have updated my answer slightly. Is it practical for you to solve your problem numerically? – lemon Nov 12 '17 at 12:41