Velocity

Displacement

Definition: Displacement

Let $\mathcal{R}$ be a reference frame with origin $O$.

The displacement of a particle in the interval $\Delta t$ between the times $t_1$ and $t_2>t_1$ is the difference in its position at $t_2$ and its position at $t_1$:

$$\mathbf{r}(t_2)-\mathbf{r}(t_1)$$

NOTATION

$$\Delta \mathbf{r}\Delta \mathbf{r}\Delta \vec{r}$$

Average Velocity

Definition: Average Velocity

Let $\mathcal{R}$ be a reference frame with origin $O$.

The average velocity of a particle during the time interval $\Delta t$ is its displacement during $\Delta t$ divided by the length of the interval:

$$\frac{\Delta \mathbf{r}}{\Delta t}$$

NOTATION

$${\mathbf{v}}_{\text{avg}}{\mathbf{v}}_{\text{avg}}{\vec{v}}_{\text{avg}}$$

Instantaneous Velocity

Definition: Instantaneous Velocity

The instantaneous velocity of a point mass at a moment $t_0$ is its average velocity within an infinitely small time interval after $t_0$:

$\mathbf{v}(t_0)\stackrel{\text{def}}{=}{\lim }_{\Delta t\to 0}{\mathbf{v}}_{\text{avg}}={\lim }_{\Delta t\to 0}\frac{\mathbf{r}(t_0+\Delta t)-\mathbf{r}(t_0)}{\Delta t}=\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}(t_0)=\dot{\mathbf{r}}(t_0)$

NOTE

The instantaneous velocity is just the temporal derivative of the position.

NOTE

When one says “velocity”, they usually mean the instantaneous velocity.

Speed

Definition: (Average) Speed

The (average) speed of a particle is the magnitude of its (average) velocity.

Bibliography