PNC-01-070 – Instantaneous Velocity

Instantaneous Velocity

Instantaneous velocity for one-dimensional motion:

The average velocity vector describes 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 because the information between time $t_i$ and $t_f=t_i+\mathrm{\Delta }t$ is not recorded. To define the instantaneous velocity, the time interval $\mathrm{\Delta }t$ between two consecutives measurements must shrink to an infinitely small amount thus providing, in a sense, a video of the particle’s motion rather than a sequence of snapshots.

We define the instantaneous velocity as the velocity of an object at any instant $t$ and we denote it by $v_x\left(t\right)$. Mathematically, we achieve this by making the time interval $\mathrm{\Delta }t$ in the average velocity formula infinitely small (shrinking it to zero without every reaching zero) which turns the average velocity into the instantaneous velocity.

Properties of instantaneous velocity:

• 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.