How many derivatives of an object's position with respect to time have to be continuous?

To determine the position, velocity, and acceleration of an object as functions of time, especially under constant acceleration, you need to consider the continuity and differentiability of the position function up to the second derivative.

1. Position ((s(t))): This is the primary function.
2. Velocity ((v(t))): This is the first derivative of the position function with respect to time, i.e., (v(t) = \frac{ds}{dt})45.
3. Acceleration ((a(t))): This is the second derivative of the position function with respect to time, i.e., (a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2})45.

For the typical analysis of uniformly accelerated motion, the position function needs to be twice differentiable, meaning it must have continuous first and second derivatives. This ensures that both the velocity and acceleration are well-defined and continuous functions of time34.