a) Not possible
b) Yes, it's possible (see graph in attachment)
Explanation:
a)
The average velocity of a body is defined as the ratio between the displacement and the time elapsed:
where
is the displacement
is the time elapsed
In this problem, we want to have zero displacement and non-zero average velocity. From the equation above, we see that this is not possible. In fact, if the total displacement is zero,
And therefore as a consequence,
which means that the average velocity is zero.
B)
Here we want to have zero displacement and non-zero velocity. In this case, it is possible: in fact, we are not talking about average velocity, but we are talking about (instantaneous) velocity.
On a position-time graph, the instantaneous velocity is the slope of the graph. Look at the graph in attachment. We see that the position of the object first increases towards positive value, then it decreases (the object starts moving backward), then becomes negative, then it increases again until returning to the original position, x = 0.
In all of this, we notice that the total displacement of the object is zero:
However, we notice that the instantaneous velocity of the object at the various instants is not zero, because the slope of the graph is not zero.
Learn more about average velocity:
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