# Is initial velocity and final velocity instantaneous?

Nov 4, 2016

Yes, correct

#### Explanation:

If displacement of a body is dependent on time and is written as $\vec{r} \left(t\right)$, its velocity $\vec{v} \left(t\right)$ is defined as first differential of displacement vector with respect to time
$\vec{v} \left(t\right) \equiv \frac{d}{\mathrm{dt}} \vec{r} \left(t\right)$

For sake of simplicity, many times we don't put the arrow marks at the top of these quantities but always understand these as vectors.

Now initial velocity is velocity of the body at time $t = {t}_{\text{initial}}$
and final velocity is velocity of thebody at time $t = {t}_{\text{final}}$

These can also be written as
Initial Velocity$= \vec{v} \left({t}_{\text{initial}}\right)$
and
Final Velocity$= \vec{v} \left({t}_{\text{final}}\right)$

It may be appreciated that at any time $t = {t}_{\text{initial}} + \delta t$
Value of velocity is given as
$= \vec{v} \left({t}_{\text{initial}} + \delta t\right)$
where $\delta t$ is infinitesimal change in time from initial condition.

This velocity could be different from $\vec{v} \left({t}_{\text{initial}}\right)$ depending upon dependence of velocity function on time $t$

Similarly, the discussion holds for final velocity.

Please feel free to raise any questions.