Is rate of change of velocity wrt distance and rate of change of velocity wrt time the same thing?

Question

Is rate of change of velocity wrt distance and rate of change of velocity wrt time the same thing?If both are same can we define acceleration in the former way?

Please explain using calculus.

Answer 1

No they aren't. Suppose we have some velocity $v(t)$. The differential with respect to time is just the acceleration:

$$ \frac{d}{dt}v(t) = a(t) $$

Now differentiate it with respect to distance $s$:

$$ \frac{d}{ds} v(t) = \frac{dt}{ds}\frac{d}{dt} v(t) = \frac{dt}{ds} a(t) = \frac{a(t)}{v(t)} $$