Answer:
no (and yes)
Step-by-step explanation:
You want to know if there's a time interval during which a particle is not moving if its position is defined by y = 0.20 +8.0t −10t².
Polynomial function
When the position is described by a polynomial function, the velocity is described by the derivative of that function. If the polynomial function describing position has a degree greater than 0, the derivative will have a degree of 0 or greater, but will not be zero except possibly at specific points.
There will be no non-zero-length intervals in which the motion is identically zero for the duration of the interval.
Given function
The position versus time graph of the given function is shown in the attachment. At the point t = 0.4, the position is unchanging. That is, on the zero-length interval [0.4, 0.4], the particle is not moving. (However, its acceleration is non-zero.) For any other interval, the particle will be in motion.
Average motion
The average motion will be zero over any time interval that is symmetric about t=0.4. For example, the position at t=0.3 and at t=0.5 is the same: y = 1.7. From one end of the interval to the other, there is no net displacement, however, the particle is moving at all points in time (except t=0.4) within that interval.
There is only a zero-length interval in which the particle is not moving. There are an infinite number of intervals in which the particle has no net motion.
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