P01-070 – Instantaneous Velocity

Instantaneous Velocity

Instantaneous velocity for one-dimensional motion:

The average velocity vector gives some information regarding how fast the particle moves, on average, over a finite amount of time. However, it fails to adequately describe the velocity of the particle at any given instant as doing so requires letting $\mathrm{\Delta }t\to 0$. Indeed, this limit shrinks the time intervals between two consecutives measurements to an infinitely small amount thus providing, in a sense, a video of the particle’s motion rather than a sequence of snapshots.

With help from calculus, we define the instantaneous velocity as the following limit

\boxed{v_x={\mathop{\mathrm{lim}}_{\mathrm{\Delta }t\to 0} \frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}\ }=\frac{dx}{dt}\ \ \ \ \ \ \ \ (m/s)}

where such a limit is, by definition, equal to the derivative of $x$ with respect to $t$ and is denoted by $\displaystyle{\frac{dx}{dt}}$.

• A positive instantaneous velocity at an instant $t$ denotes motion in the positive $x$ direction at that given instant.

• A negative instantaneous velocity at an instant $t$ denotes motion in the negative $x$ direction at that given instant.

Example: if the position $x$ of a particle as it moves along a given path is described by the function $x\left(t\right)=3t^2-t+2$, then we can conclude that its instantaneous velocity is equal to

v_x\left(t\right)=\frac{dx}{dt}=6t-1\ \ \ \ \ (m/s)

This yields a function of time for $v_x\left(t\right)$ which will output the instantaneous velocity of the particle at any instant $t$ of its motion.