P01-180 – Instantaneous Velocity in 3D

Instantaneous Velocity in Three Dimensions

We previously defined the instantaneous velocity vector of a point moving on the $x$-axis as

v_x={\mathop{\mathrm{lim}}_{\mathrm{\Delta }t\to 0} \frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}\ }=\frac{dx}{dt}

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.

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

\boxed{\overrightarrow{v}\left(t\right)=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}}

Notation: the components of the instantaneous velocity vector are typically denoted by

\boxed{v_x=\frac{dx}{dt}} \ \ \ \ \ and \ \ \ \ \ \boxed{v_y=\frac{dy}{dt}}

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

\boxed{\overrightarrow{v}\left(t\right)=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}}

Notation: the components of the instantaneous velocity vector are typically denoted by

\boxed{v_x=\frac{dx}{dt}} \ \ \ \ \ and \ \ \ \ \ \boxed{v_y=\frac{dy}{dt}} \ \ \ \ \ and\ \ \ \ \ \boxed{v_z=\frac{dz}{dt}}

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

Example: a particle has a position vector given by $\overrightarrow{r}\left(t\right)=\left(t^2-1\right)\hat{x}+\left(7t+2\right)\hat{y}+\left(t^2+3t\right)\hat{z}$. What is its velocity vector?

The velocity vector of this particle is equal to

\overrightarrow{v}\left(t\right)=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}=2t\ \hat{x}+7\hat{y}+\left(2t+3\right)\hat{z}

Note: while there is nothing fundamentally wrong with the above notation, it is more customary to track the components of the velocity separately as follows (rather than packed into one single vector).

\begin{aligned}
\left\{ \begin{array}{c}
x\left(t\right)=t^2-1 \ \\ 
y\left(t\right)=7t+2 \\\ 
z\left(t\right)=t^2+3t \end{array}
\right.\ \ \ \ \Rightarrow \ \ \ \ \ \left\{ \begin{array}{c}
v_x\left(t\right)=2t \ \ \ \ \ \ \  \\ 
v_y\left(t\right)=7 \ \ \ \ \ \ \ \ \\ 
v_z\left(t\right)=2t+3 \end{array}
\right.
\end{aligned}