[Intro Dynamics] How to obtain v(t) from v(r) in 3D?

[Leo Liu]

Homework Statement

Conceptual question

Relevant Equations

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We know that in 1D, we can perform the following steps to obtain velocity in terms of time if the velocity is a function of position:
$$v=f(x)$$
$$dt=\frac{dx}{v}$$
$$t_2-t_1=\int_{x_1}^{x_2}\frac{dx}{v}$$
$$x(t)\rightarrow v(t)$$

But I wonder if it is possible to obtain $\vec v(t)$ from $\vec v(\vec r)$? Does it even make sense to do something like $$\Delta t=\int_{r_{xi}}^{r_{xf}} \frac{d r_{x}}{v(r_x,r_y,r_z)}$$?

 

[anuttarasammyak]

The formula for time
$$t_{01}=\int_{l_0}^{l_1} \frac{dl}{v(l)}$$
where l is 2d or 3d trajectory of the particle whose position and speed are defined as function of l , would be of your interest.