Physical interpretation of integral of function of distance

Question

I wrote the equation for the time it takes for an object to travel a specific distance at a specific speed. This equation is defined as $t(s) = \frac cs$ where $d$ is a constant for the distance the object will travel where $s$ is the speed at which the object is traveling.

Likewise, this is a simple rational function where the faster the object is traveling, the less time it will take for that object to reach the distance defined by $d$. However, I took the indefinite integral of this function which was equal to $d\ln|s|+C$.

I was wondering what the interpretation of this integral would be in regard to physics. I believe it could be the sum of the time that the object is moving at that specific speed, but I'm not sure if there is a physical representation of this integral. Also, sorry for my inexperience in MathJax.

Answer 1

I was wondering what the interpretation of this integral would be in regard to physics.

I don't think your integral has any physical significance. This is because when you use the equation $t=\frac cs$ where $t$ is the time it takes to travel a distance of $c$ at a speed $s$, you are already assuming a constant $s$. Therefore, if you decide to integrate this equation with respect to $s$, you are considering a varying speed mathematically, which then takes you out of the physical interpretation of the equation.

I believe it could be the sum of the time that the object is moving at that specific speed

If you are integrating with respect to the speed $s$, then your integral has units of distance, not of time. Therefore this cannot be the interpretation of the integral, even if it did have physical significance.

Answer 2

We have

$$\int_{s_0}^{s} t(s)\,ds=\int_{t_0}^{t} t\frac{ds(t)}{dt}dt$$

and by parts

$$\int_{t_0}^{t} t\frac{ds(t)}{dt}dt=\left.ts\right|_{t_0}^t-\int_{t_0}^t s(t)\,dt=ts(t)-t_0s_0-x(t)+x_0$$ where $x$ is the distance traversed.

The terms have the dimension of a distance. The first one doesn't have a common interpretation. It can be seen as "the traversed distance, had the speed kept the current value all the time".