Does Euler number $e$ have a role in kinematics?

A simple example is just an object starting at rest falling with a drag force proportional to the velocity of the object, $F_D=-bv$. Then the acceleration is given by

$$a=\frac{dv}{dt}=g-\frac bmv$$

Therefore, the velocity over time is given by $$v(t)=\frac{mg}{b}(1-e^{-bt/m})$$

Typically you get $e$ popping up when the rate of change of something is proportional to itself.

Of course, $e$ is ubiquitous in kinematics. For example, consider a repulsive force proportional to $x$, $$F = kx.$$ Then the acceleration is $$a = \frac{k}{m} x = \omega^2 x, \quad \omega = \sqrt{\frac{k}{m}}.$$ This differential equation has solutions of the form $e^{\omega t}$ and $e^{- \omega t}$. In particular, suppose that $x(0) = 1$ and $v(0) = \omega$. In that case the solution is exactly $$x(t) = e^{\omega t}.$$ In general, for any linear force law, the solutions will be exponentials or complex exponentials, so it's honestly hard to avoid using $e$.