PNC-01-180 – Instantaneous Velocity in 3D

Instantaneous Velocity in Three Dimensions

We previously defined the instantaneous velocity $v_x\left(t\right)$ of a point moving on the $x$-axis and we now extend this definition to two-dimensional motion and three-dimensional motion by adding the corresponding components along the $y$-direction and $z$-direction respectively.

Instantaneous velocity vector in a 2D coordinate system:

Consider a particle with a position vector $\overrightarrow{r}=x\left(t\right)\hat{x}+y\left(t\right)\hat{y}$.

The instantaneous velocity $\overrightarrow{v}\left(t\right)$ of this particle is given by the vector

\boxed{\overrightarrow{v}\left(t\right)=v_x\ \hat{x}+v_y\ \hat{y}}

where the components $v_x$ and $v_y$ are the instantaneous velocities in the $x$ and $y$ directions respectively.

Instantaneous velocity vector in a 3D coordinate system:

Consider a particle with a position vector $\overrightarrow{r}=x\left(t\right)\hat{x}+y\left(t\right)\hat{y}+z\left(t\right)\hat{z}$.

The instantaneous velocity $\overrightarrow{v}\left(t\right)$ of this particle is given by the vector

\boxed{\overrightarrow{v}\left(t\right)=v_x\ \hat{x}+v_y\ \hat{y}+v_z\ \hat{z}}

where the components $v_x$, $v_y$, and $v_z$ are the instantaneous velocities in the $x$, $y$, and $z$ directions respectively.

Note: it is common to simply call the instantaneous velocity vector of a particle its velocity vector or, simply, its velocity.