# Blog & News

Average velocity: is defined as the displacement vector divided by the total time from start to finish.

\[v_{av-x}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}\]

Instantaneous velocity: is the limit of the average velocity as $\Delta t$ approaches zero. In one dimension, say $x$, is defined by

\[v_x=\lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t}=\frac{dx}{dt}\]

Instantaneous acceleration: is the limit of the average acceleration as $\Delta t$ approaches zero. In one dimension, say $x$, is defined by

\[a_x=\lim_{\Delta t\to 0}\frac{\Delta v_x}{\Delta t}=\frac{dv_x}{dt}=\frac{d^2 x}{dt^2}\]

$d^{2}x/dt^2$ is the second derivative of the position vector function with respect to time. The instantaneous acceleration is the slope of the $v_x$-versus-time graph.

If the magnitude of the velocity is increasing, the motion is speeding up. This case occurs when $a_x$ and $v_x$ have the same sign. But if $a_x$ and $v_x$ have opposite signs the motion is slowing down or the magnitude of velocity of the motion is decreasing.

The slope of the line $AB$ in the position-versus-time graph indicates the average velocity of an object.

In the following graph, if the time interval $\Delta t$ becomes smaller and smaller, the point $B$ moves closer and closer to the point $A$ and finally the straight line $AB$ at point $A$ becomes tangent to the curve. In such a case, the line’s slope at time $t_1$ represents the instantaneous velocity.