Question

During which time interval are particles described by these position graphs at rest?

Answer 1

We know that a given particle is at rest when its speed is zero. Hence if we determine the time interval where the speed of the body is zero we will know the time interval where it is at rest. In the distance -time graph speed is given by its slope.

Complete answer:
A particle's speed is zero while it is at rest. And because speed is the slope of the x-t graph, it is evident that the graph becomes horizontal (slope =0) at the interval of $2 - 3s.$As a result, this interval is the solution.

Additional information:
The distance time graph places and discusses three linked components: time, distance, and speed. Here,
The bottom X-axis represents time.
The vertical Y-axis is used to represent distance.
The distance vs. time graph depicts many changes in an object's movement. The straight line on a graph, for example, indicates that an item moves at a constant pace(speed). If we see the above given graph constant speed is traveled in the time interval 0-2 seconds. Similarly there are a few more conclusions that can be drawn, they are :The speed is growing or accelerating when a curve is upward and when the curve is downward, however, it indicates that an object's motion is slowing down(decelerating).

Note:
Remember that only the slope of the distance-time graph gives speed. The slope being zero in velocity-time graphs or acceleration-time graphs doesn’t signify that the body is at rest.