I put to you that they wouldn't typically introduce even velocities and accelerations as being time derivatives of position. You just have to accept:
x(t) = x0 + v0t + (1/2)at2
Actually, even introducting the idea of "time-derivatives of positions" isn't that demanding, as long as you don't insist that students be able to manipulate the derivatives themselves. Calculus is easy conceptually if you don't need to work the damn sums yourself.
Define the concepts of "velocity" and "acceleration" in terms of "rates of change" -- you don't even need the term "derivative." The experiment you define above can be demonstrated empirically to justify using it. From there, you've got everything you need.
), even though these are observable physical phenomena.