P01-170 – Average Acceleration Vector in 3D

Average Acceleration in Three Dimensions

We previously defined the average acceleration vector of a point moving on the $x$-axis with velocity ${\overrightarrow{v}}_i=v_{ix}\ \hat{x}$ at time $t_i$ and a velocity ${\overrightarrow{v}}_f=v_{fx}\ \hat{x}$ at time $t_f$ as

{\overrightarrow{a}}{avg\ x}=\frac{\mathrm{\Delta }\overrightarrow{v}}{\mathrm{\Delta }t}=\frac{{\overrightarrow{v}}_f-{\overrightarrow{v}}_i}{t_f-t_i}=\frac{v_{fx}-v_{ix}}{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 acceleration vector in a 2D coordinate system:

Consider a particle with a velocity ${\overrightarrow{v}}_i=v_{ix} \ \hat{x}+v_{iy} \ \hat{y}$ at time $t_i$ and a velocity ${\overrightarrow{v}}_f=v_{fx} \ \hat{x}+v_{fy} \ \hat{y}$ at time $t_f$. The average acceleration vector of this particle is then defined by

\boxed{{\overrightarrow{a}}{avg}=\frac{\mathrm{\Delta }v_x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }v_y}{\mathrm{\Delta }t}\ \hat{y}=\frac{v_{fx}-v_{ix}}{t_f-t_i}\ \hat{x}+\frac{v_{fy}-v_{iy}}{t_f-t_i}\ \hat{y}}

Notation: the components of the average velocity vector are

\boxed{a_{avg\ x}=\frac{\mathrm{\Delta }v_x}{\mathrm{\Delta }t}}\ \ \ \ \ and\ \ \ \ \ \boxed{a_{avg\ y}=\frac{\mathrm{\Delta }v_y}{\mathrm{\Delta }t}}

Average acceleration vector in a 3D coordinate system:

Consider a particle with a velocity ${\overrightarrow{v}}_i=v_{ix} \ \hat{x}+v_{iy} \ \hat{y}+v_{iz} \ \hat{z}$ at time $t_i$ and a velocity ${\overrightarrow{v}}_f=v_{fx} \ \hat{x}+v_{fy} \ \hat{y}+v_{iz} \ \hat{z}$ at time $t_f$. The average acceleration vector of this particle is then defined by

\boxed{{\overrightarrow{a}}{avg}=\frac{\mathrm{\Delta }v_x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }v_y}{\mathrm{\Delta }t}\ \hat{y}+\frac{\mathrm{\Delta }v_z}{\mathrm{\Delta }t}\ \hat{z}=\frac{v_{fx}-v_{ix}}{t_f-t_i}\ \hat{x}+\frac{v_{fy}-v_{iy}}{t_f-t_i}\ \hat{y}+\frac{v_{fz}-v_{iz}}{t_f-t_i}\ \hat{z}}

Notation: the components of the average velocity vector are

\boxed{a_{avg\ x}=\frac{\mathrm{\Delta }v_x}{\mathrm{\Delta }t}} \ \ \ \ \ and \ \ \ \ \ \boxed{a_{avg\ y}=\frac{\mathrm{\Delta }v_y}{\mathrm{\Delta }t}} \ \ \ \ \ and \ \ \ \ \ \boxed{a_{avg\ z}=\frac{\mathrm{\Delta }v_z}{\mathrm{\Delta }t}}

Example: a particle with velocity ${\overrightarrow{v}}_i=4\hat{x}+2\hat{y}-\hat{z}$ sees its velocity change to ${\overrightarrow{v}}_f=-2\hat{x}+3\hat{z}$ in time $\mathrm{\Delta }t=2\ s$. What is its average acceleration vector?

Its average acceleration vector is then equal to

{\overrightarrow{a}}_{avg}=\frac{\mathrm{\Delta }v_x}{\mathrm{\Delta }t}\ \hat{x}+\frac{\mathrm{\Delta }v_y}{\mathrm{\Delta }t}\ \hat{y}+\frac{\mathrm{\Delta }v_z}{\mathrm{\Delta }t}\ \hat{z}=\frac{-2-4}{2}\hat{x}+\frac{0-2}{2}\hat{y}+\frac{3-\left(-1\right)}{2}\hat{z}=-3\hat{x}-\hat{y}+2\hat{z}

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

\left\{ \begin{array}{c}
a_{avg\ x}=-3 \\
a_{avg\ y}=-1 \\
a_{avg\ z}= \ \ \ 2 \end{array}
\right.