PNC-01-190 – Instantaneous Acceleration in 3D

Instantaneous Acceleration in Three Dimensions

We previously defined the instantaneous acceleration $a_x\left(t\right)$ 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 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 the vector

\boxed{\overrightarrow{a}\left(t\right)=a_x\ \hat{x}+a_y\ \hat{y}}

where the components $a_x$ and $a_y$ are the instantaneous accelerations in the $x$ and $y$ directions respectively.

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 the vector

\boxed{\overrightarrow{a}\left(t\right)=a_x\hat{x}+a_y\hat{y}+a_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 acceleration vector of a particle its acceleration vector or, simply, its acceleration.