Average Velocity

For a particle confined to move along the $x$-direction, average velocity is just the change in the $x$-coordinate divided by the elapsed time:

Average velocity remains a vector quantity even in one dimension, but with only two possible directions (positive $x$ or negative $x$) along a line, a plain $+$ or $-$ sign is enough to record direction — no separate arrow notation is needed.

The instantaneous velocity is, as before, the limit of this ratio as the time interval shrinks to zero:

Graphically, the slope of the position-time ($x$-$t$) graph at a point gives the velocity at that instant.

Going the other way — from a velocity-time graph back to displacement. Since $v=dx/dt$, we can write $dx = v\,dt$, and integrating both sides between times $t_1$ and $t_2$,

$\int_{x_1}^{x_2} dx = \int_{t_1}^{t_2} v\,dt \qquad\Longrightarrow\qquad x_2-x_1 = \int_{t_1}^{t_2} v\,dt$ …