# How do you calculate instantaneous speed?

Apr 2, 2016

$| \vec{v} | \left({t}_{1}\right) = | \frac{d}{\mathrm{dt}} \vec{x} \left(t\right) {|}_{{t}_{1}}$

#### Explanation:

If $\vec{x} \left(t\right)$ is the position vector, which is a function of time, of an object, then the velocity vector $\vec{v}$ in general is defined as the first derivative of the position vector with respect to time.

Mathematically:
$\vec{v} \left(t\right) \equiv \frac{d}{\mathrm{dt}} \vec{x} \left(t\right)$
Nature of Velocity vector with respect to time is dictated by the nature of position vector.
To find the instantaneous speed we need to find the instantaneous magnitude of the position vector at the desired time.

Suppose we want to know the instantaneous speed at time $t = {t}_{1}$. It can be found as:
$| \vec{v} | \left({t}_{1}\right) = | \frac{d}{\mathrm{dt}} \vec{x} {|}_{{t}_{1}}$