The displacement-time graph for a particle travelling along the...

The slope of a displacement-time ($x-t$) graph represents the velocity of the particle, $v = \dfrac{dx}{dt}$.

For the time interval $0 \le t \le t_0$, the graph is a straight line with a constant positive slope. This indicates that the particle is travelling at a constant positive velocity.

For the time interval $t > t_0$, the graph is a horizontal straight line. The slope of a horizontal line is zero, which means the velocity of the particle is zero. Therefore, the particle comes to rest (stops) after time $t_0$.

Thus, the particle travels at a constant velocity up to time $t_0$, and subsequently stops.