Velocity

Velocity is a rate of change for position per unit of time, and unlike speed, it requires a direction.

100 kilometers per hour, north

Average Velocity

v = \dfrac {\Delta x} {\Delta t}

Instantaneous Velocity

v = \dfrac {dx} {dt}

v dt = dx

\int_{t_i}^{t_f} v dt = \int_{x_i}^{x_f} dx

v (t_f - t_i) = x_f - x_i

In a way, I want to stop here. What we have above is correct, and it has everything shown in it. What happens is, we can make some assumptions have some things go to zero, and that makes it simpler.

It is typical for authors to drop the f subscript and just show the variable, in fact this is done so much we want you to be familiar with it. They also frequently use a subscript of 0 where we used an i.

Almost always time equals 0 at the start of the experiment so t_i=0 and any term with it disappears.

v (t_f ) = x_f - x_i

x_f = v (t_f) + x_i

x= v (t) + x_0