1. The problem asks to determine the time $t$ when the rate of change of the position $p$ of a weather balloon is undefined based on the given graph. 2. The rate of change of position with respect to time is the derivative $\frac{dp}{dt}$, which corresponds to the slope of the graph at each point. 3. The rate of change is undefined where the graph has vertical segments or sharp vertical drops because the slope is infinite or does not exist there. 4. From the graph description, vertical drops occur at $t=2$ and $t=8$. 5. Therefore, the times when the rate of change is undefined are $t=2$ and $t=8$. **Final answer:** $$t=2 \text{ and } t=8$$