P01-160 – Average Velocity Vector in 3D
Average Velocity in Three Dimensions
We previously defined the average velocity vector of a point moving on the $x$-axis as
{\overrightarrow{v}}_{avg\ x}=\frac{\mathrm{\Delta }\overrightarrow{r}}{\mathrm{\Delta }t}=\frac{{\overrightarrow{r}}_f-{\overrightarrow{r}}_i}{t_f-t_i}=\frac{x_f-x_i}{t_f-t_i}\ \hat{x}This definition can be extended to two-dimensional motion and three-dimensional motion by adding the corresponding components along the $y$-direction and $z$-direction respectively.
Average velocity vector in a 2D coordinate system:
Consider a particle that undergoes, in an amount of time $\mathrm{\Delta }t$ a displacement $\mathrm{\Delta }\overrightarrow{r}$ equal to
\mathrm{\Delta }\overrightarrow{r}=\left(x_f-x_i\right) \ \hat{x}+\left(y_f-y_i\right) \ \hat{y}The average velocity vector of this particle is then defined by
\boxed{{\overrightarrow{v}}_{avg}=\frac{\mathrm{\Delta }\overrightarrow{r}}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}\ \hat{y}}Notation: the components of the average velocity vector are
\boxed{v_{avg\ x}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}}\ \ \ \ \ and\ \ \ \ \ \boxed{v_{avg\ y}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}}Average velocity vector in a 3D coordinate system:
Consider a particle that undergoes, in an amount of time $\mathrm{\Delta }t$ a displacement $\mathrm{\Delta }\overrightarrow{r}$ equal to
\mathrm{\Delta }\overrightarrow{r}=\left(x_f-x_i\right)\ \hat{x}+\left(y_f-y_i\right)\hat{y}+\left(z_f-z_i\right)\hat{z}The average velocity vector of this particle is then defined by
\boxed{{\overrightarrow{v}}_{avg}=\frac{\mathrm{\Delta }\overrightarrow{r}}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}\ \hat{y}+\frac{\mathrm{\Delta }z}{\mathrm{\Delta }t}\ \hat{z}}Notation: the components of the average velocity vector are
\boxed{v_{avg\ x}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}}\ \ \ \ \ and \ \ \ \ \ \boxed{v_{avg\ y}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}} \ \ \ \ \ and \ \ \ \ \ \boxed{v_{avg\ z}=\frac{\mathrm{\Delta }z}{\mathrm{\Delta }t}}Example: a particle located at ${\overrightarrow{r}}_i=2\hat{x}-\hat{y}+6\hat{z}$ undergoes a displacement $\mathrm{\Delta }\overrightarrow{r}$ that bring it to a position ${\overrightarrow{r}}_f=-2\hat{x}+3\hat{y}$ in time $\mathrm{\Delta }t=2\ s$. What is its average velocity vector?
The displacement vector of the particle is equal to
\mathrm{\Delta }\overrightarrow{r}={\overrightarrow{r}}_f-{\overrightarrow{r}}_i=\left(-2-2\right)\hat{x}+\left(3-\left(-1\right)\right)\hat{y}+\left(0-6\right)\hat{z}=-4\hat{x}+4\hat{y}-6\hat{z}Its average velocity vector is then equal to
{\overrightarrow{v}}_{avg}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }y}{\mathrm{\Delta }t}\ \hat{y}+\frac{\mathrm{\Delta }z}{\mathrm{\Delta }t}\ \hat{z}=-\frac{4}{2}\hat{x}+\frac{4}{2}\hat{y}-\frac{6}{2}\hat{z}=-2\hat{x}+2\hat{y}-3\hat{z}Note: while there is nothing fundamentally wrong with the above notation, it is more customary to track the components of the average velocity separately as follows (rather than packed into one single vector).
\left\{ \begin{array}{c}
v_{avg\ x}=-2 \ \\
v_{avg\ y}=2 \ \ \ \ \\
v_{avg\ z}=-3 \ \end{array}
\right.