What would I get if I differentiated the path of a particle?

Question

While I was reading about motion with an emphasis on vectors I noticed that if we took the displacement vector from any point to another, (that is $\vec \Delta r = \vec r2 - \vec r1$ where $\vec r2$ is the position vector of the end point and $\vec r1$ is the position vector of the start point), we get something like a secant line. Now I thought perhaps to get the distance instead of the vector, we could differentiate the curve then integrate it from wherever to wherever. Another way of putting it is we make the time intervals over which we take $\vec \Delta r$ so small that $\vec \Delta r$ eventually traces the path of the particle instead of giving a secant line. Then we could add all these $\Delta r$'s together. Basically that would be like differentiating it then integrating it. But I'm not sure about this because I don't know what value the y-axis takes on because I'm not sure how to define "the path of a particle" in numerical values like meters etc. So I'd really like it if someone could tell me what the y-axis is in this case (x is time obviously) and what we would get upon differentiating it, and if the method I outlined to calculate the distance of the particle (differentiating then integrating) makes sense. If it is wrong, I'd like it if you could tell me exactly how to calculate the total distance travelled by a particle from A to B, given only the path of the particle.

Answer 1

The setup I think you're discussing is the idea that if we consider a particle with position $(x_1(t), x_2(t), x_3(t))$ at time $t$, then integrating $(\dot x_1, \dot x_2, \dot x_3)$ should give you $x$. That's true: $\int_{t_0}^{t_1} \dot x_i = x_i(t_1) - x_i(t_0)$, which is precisely the displacement along the $x_i$ axis between time $t_0$ and $t_1$. (If the multidimensional setup confuses you, just consider a point travelling on the real line.) That isn't the same as the distance travelled by the particle, though, unless it's moving along a straight line. For that, you want the length $L$ of the path $(x_i(t))$ from time $t_0$ to $t_1$, which is given by \begin{align*} L = \int_{t_0}^{t_1} dt\, \sqrt{\dot x_1^2 + \dot x_2^2 + \dot x_3^2} \end{align*} To derive that equation of $L$, divide $x$ into small line segments and note that the displacement of $x$ after a short time $\Delta t$ is approximately $(\dot x_1, \dot x_2, \dot x_3)\Delta t$. (There are some technical considerations I'm sweeping under the rug, but since this question was asked in the context of mechanics rather than analysis, I'm going to happily ignore them.)