There is a question regarding basic physical understanding. Assume you have a mass point (or just a ball if you like) that is constrained on a line. You know that at $t=0$ its position is $0$, i.e., $x(t=0)=0$, same for its velocity, i.e., $\dot{x}(t=0)=0$, its acceleration, $\ddot{x}(t=0)=0$, its rate of change of acceleration, $\dddot{x}(t=0)=0$, and so on. Mathematically, for the trajectory of the mass point one has
\begin{equation} \left. \frac{d^{n}x}{dt^n}\right|_{t=0} = 0 \textrm{ for } n \in \mathbb{N}_0\mbox{.} \end{equation}
My physical intuition is that the mass point is not going to move because at the initial time it had no velocity, acceleration, rate of change of acceleration, and so on. But the mass point not moving means that $x(t) \equiv 0$ since its initial position is also zero. However, it could be that the trajectory of the mass point is given by $x(t) = \exp(-1/t^2)$. This function, together with all its derivatives, is $0$ at $t=0$ but is not equivalent to zero. I know that this function is just not analytical at $t=0$. My question is about the physical understanding: How could it be that at a certain moment of time the mass point has neither velocity, nor acceleration, nor rate of change of acceleration, nor anything else but still moves?
No comments:
Post a Comment