P01-190 – Instantaneous Acceleration in 3D

Instantaneous Acceleration in Three Dimensions

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

a_x={\mathop{\mathrm{lim}}_{\mathrm{\Delta }t\to 0} \frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}\ }=\frac{dv_x}{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 acceleration vector in a 2D coordinate system:

Consider a particle with a velocity vector $\overrightarrow{v}=v_x\left(t\right)\hat{x}+v_y\left(t\right)\hat{y}$.

The instantaneous acceleration $\overrightarrow{a}\left(t\right)$ of this particle is given by

\boxed{\overrightarrow{a}\left(t\right)=\frac{dv_x}{dt}\hat{x}+\frac{dv_y}{dt}\hat{y}}

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

\boxed{a_x=\frac{dv_x}{dt}} \ \ \ \ \ and \ \ \ \ \ \boxed{a_y=\frac{dv_y}{dt}}

Instantaneous acceleration vector in a 3D coordinate system:

Consider a particle with a velocity vector $\overrightarrow{v}=v_x\left(t\right)\hat{x}+v_y\left(t\right)\hat{y}+v_z\left(t\right)\hat{z}$.

The instantaneous acceleration $\overrightarrow{a}\left(t\right)$ of this particle is given by

\boxed{\overrightarrow{a}\left(t\right)=\frac{dv_x}{dt}\hat{x}+\frac{dv_y}{dt}\hat{y}+\frac{dv_z}{dt}\hat{z}}

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

\boxed{a_x=\frac{dv_x}{dt}} \ \ \ \ \ and \ \ \ \ \ \boxed{a_y=\frac{dv_y}{dt}} \ \ \ \ \ and \ \ \ \ \ \boxed{a_z=\frac{dv_z}{dt}}

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

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

The acceleration vector of this particle is equal to

\overrightarrow{a}\left(t\right)=\frac{dv_x}{dt}\hat{x}+\frac{dv_y}{dt}\hat{y}+\frac{dv_z}{dt}\hat{z}=2\hat{x}+2\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).

\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.\ \ \ \ \Rightarrow \ \ \ \ \ \left\{ \begin{array}{c}
a_x\left(t\right)=2 \\ 
a_y\left(t\right)=0 \\ 
a_z\left(t\right)=2 \end{array}
\right.