What does the derivative of velocity with respect to position mean?

Question

According to a Physics book, for a particle undergoing motion in one dimension (like a ball in free fall) it follows that

$$\frac{dv}{ds} = \frac{dv}{dt} \frac{dt}{ds} = \frac{a}{v},$$

where $v$ is the velocity and $s$ is the position of the particle.

But I have problems understanding this, specially because of the use of Leibniz's notation.

I think of the position of a particle at time $t$ (under a frame of reference) as described by the image of a function $s$. The number $s(t)$ represents the particle's 'coordinate on the axis' at time $t$. Then the function $s$ relates instants of time to those points in space where the particle is supposed to be.

The velocity of the particle at time $t$ is then $s'(t)$ and the acceleration $s''(t)$. We denote the function $s'$ simply as $v$ and $s''$ as $a$.

It is supposed that using the chain rule yields the previous the result of the book, but formally the chain rule is stated as

$$(f \circ g)' = (f' \circ g) \cdot g'$$

for any two differentiable functions $f$ and $g$.

Then:

Why is the velocity function treated as a composition?

What function does $\frac{dv}{ds}$ represent?

Answer 1

What's confusing you is that the (perfectly standard) notation sucks, because the same letter $t$ is being used both to denote a number (that is, a particular time) and a function (that is, time as a function of position).

To reduce confusion, let $t$ be the time (a number) and let $T$ be the function that tells you, for any given position, the time at which your particle achieves that position. So at any given moment $t$, if the particle is at $s$, then $t=T(s)$.

The next thing that sucks about the notation is that $v$ is being used for two different functions: Velocity as a function of time and velocity as a function of position. To clear this up, introduce two functions: $w(t)$ tells you the velocity at time $t$ and $v(s)$ tells you the velocity when the particle is at position $s$. Then

$$v(s)=w(T(s))$$

Apply the chain rule to get $$v'(s)=w'(T(s))T'(s)$$

Because the Leibniz notation freely confuses $t$ with $T$ and $v$ with $w$, $w'(T(s))$ ends up being written as $dv/dt$ and $T'(s)$ ends up being written as $dt/ds$. So that gives you your formula.

Answer 2

I think I may clear up the issue.

s(t) is not position it is the arc length function, it gives you the length a particle has moved along curve x(t) for a time interval t.

ds/dt is the instantaneous tangential speed of the particle also known as |v| or |dx/dt|.

So s(t) is the integral of instantaneous velocity with respect to time. And dv/ds is the rate of change of velocity with respect to arc length. The chain rule you posted checks out.