The figure shows the $x-t$ plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest,and in which is it the least? Give the sign of average velocity for each interval.

(N/A) $1$. Average speed is determined by the magnitude of the slope of the $x-t$ graph. The slope is given by $\frac{\Delta x}{\Delta t}$. Since the time intervals $\Delta t$ are equal,the interval with the steepest slope (greatest magnitude) has the greatest average speed,and the interval with the shallowest slope (smallest magnitude) has the least average speed.
$2$. By observing the graph:
- In interval $1$,the slope is positive and moderate.
- In interval $2$,the slope is positive and very small (the graph is nearly flat).
- In interval $3$,the slope is negative and very steep.
$3$. Comparing the magnitudes of the slopes: The magnitude of the slope is greatest in interval $3$ and least in interval $2$.
$4$. Therefore,the average speed is greatest in interval $3$ and least in interval $2$.
$5$. The sign of the average velocity corresponds to the sign of the slope:
- Interval $1$: Positive slope,so average velocity is positive.
- Interval $2$: Positive slope,so average velocity is positive.
- Interval $3$: Negative slope,so average velocity is negative.