Position, velocity, and acceleration #rkv

The two basic geometric objects we are using are positions and vectors. Positions describe locations in space, while vectors describe length and direction (no position information). To describe the kinematics (motion) of bodies we need to relate positions and vectors to each other.

Position vectors #rkv‑sp

Two positions $P$ and $Q$ can be used to define a vector $\vec{r}_{PQ} = \overrightarrow{PQ}$ from $P$ to $Q$ . We call this the relative position of $Q$ from $P$ . If we start from the origin $O$ , so we have $\vec{r}_{OP} = \overrightarrow{OP}$ , then we call this the position vector of position $P$ . When it is clear, we will write $\vec{r}_P$ for this position vector, or sometimes even just $\vec{r}$ .

Points $P$ and $Q$ and their relative and absolute position vectors. Note that we can write the position vectors with respect to different origins and in different bases. #rkv‑fp

Transformation of position vectors #rkv‑st

The position vector $\vec{r}_{OP}$ of a point $P$ depends on which origin we are using. Using a different origin will result in a different position vector for the same point. The position vectors of a point from two different origins differ by the offset vector between the origins:

Change of origin for position vectors. #rkv‑eo

$\begin{aligned} \overrightarrow{O_1 P} &= \overrightarrow{O_1 O_2} + \overrightarrow{O_2 P} \\ \vec{r}_{O_1 P} &= \vec{r}_{O_1 O_2} + \vec{r}_{O_2 P} \end{aligned}$

Position vectors are defined by the origin and the point, but not by any choice of basis. We can write any position vector in any basis and it is still the same vector.

Basis for $\vec{r}_{O_1P}$ :	none $\hat\imath,\hat\jmath$ $\hat{u},\hat{v}$
Basis for $\vec{r}_{O_2P}$ :	none $\hat\imath,\hat\jmath$ $\hat{u},\hat{v}$

Points $P$ and $Q$ and their relative and absolute position vectors. Note that we can write the position vectors with respect to different origins and in different bases, in any combination. #rkv‑fo

Velocity and acceleration vectors #rkv‑sv

The velocity $\vec{v}$ and acceleration $\vec{a}$ are the first and second derivatives of the position vector $\vec{r}$ . Technically, this is the velocity and acceleration relative to the given origin, as discussed in detail in the sections on relative motion and frames.

Definition of velocity $\vec{v}$ and acceleration $\vec{a}$ .#rkv‑ev

$\begin{aligned} \vec{v} &= \dot{\vec{r}} \\ \vec{a} &= \dot{\vec{v}} \end{aligned}$

The velocity can be decomposed into components parallel and perpendicular to the position vector, reflecting changes in the length and direction of $\vec{r}$ .

Decomposition of velocity and acceleration vectors. #rkv‑ec

$\begin{aligned} \vec{v}_\text{proj} &= \operatorname{Proj}(\vec{v}, \vec{r}) = \dot{r} \hat{r} \\ \vec{v}_\text{comp} &= \operatorname{Comp}(\vec{v}, \vec{r}) = r \dot{\hat{r}} \\ \vec{a}_\text{proj} &= \operatorname{Proj}(\vec{a}, \vec{v}) = \dot{v} \hat{v} \\ \vec{a}_\text{comp} &= \operatorname{Comp}(\vec{a}, \vec{v}) = v \dot{\hat{v}} \end{aligned}$

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Velocity and acceleration of various movements. Compare to Figure #rvc-fp. #rkv‑fa

Velocity and acceleration in Cartesian basis #rkv‑sr

Differentiating in a fixed Cartesian basis can be done by differentiating each component.

Velocity and acceleration in Cartesian basis. #rkv‑er

$\begin{aligned} \vec{r} &= r_1 \,\hat\imath + r_2 \,\hat\jmath + r_3 \,\hat{k} \\ \vec{v} &= \dot{r}_1 \,\hat\imath + \dot{r}_2 \,\hat\jmath + \dot{r}_3 \,\hat{k} \\ \vec{a} &= \ddot{r}_1 \,\hat\imath + \ddot{r}_2 \,\hat\jmath + \ddot{r}_3 \,\hat{k} \end{aligned}$

Derivation

Assuming $\hat\imath,\hat\jmath,\hat{k}$ are all fixed vectors, we can differentiate twice using #rvc-ec.

Velocity and acceleration in polar basis #rkv‑sl

Computing velocity and acceleration in a polar basis must take account of the fact that the basis vectors are not constant.

Velocity and acceleration in polar basis. #rkv‑ep

$\begin{aligned} \vec{r} &= r \,\hat{e}_r \\ \vec{v} &= \dot{r} \,\hat{e}_r + r \dot\theta \,\hat{e}_\theta \\ \vec{a} &= (\ddot{r} - r\dot\theta^2) \,\hat{e}_r + (r \ddot\theta + 2 \dot{r} \dot\theta) \,\hat{e}_\theta \end{aligned}$

Derivation

Starting from the position vector $\vec{r} = r\,\hat{e}_r$ , we differentiate and use the basis vector derivatives $\dot{\hat{e}}_r = \dot\theta \,\hat{e}_\theta$ and $\dot{\hat{e}}_\theta = -\dot\theta \,\hat{e}_r$ , giving: $\begin{aligned} \vec{r} &= r \,\hat{e}_r \\ \vec{v} &= \dot{\vec{r}} \\ &= \dot{r} \,\hat{e}_r + r \,\dot{\hat{e}}_r \\ &= \dot{r} \,\hat{e}_r + r \dot\theta \, \hat{e}_\theta \\ \vec{a} &= \dot{\vec{v}} \\ &= \ddot{r} \,\hat{e}_r + \dot{r} \,\dot{\hat{e}}_r + \dot{r} \dot\theta \,\hat{e}_\theta + r \ddot\theta \,\hat{e}_\theta + r \dot\theta \,\dot{\hat{e}}_\theta \\ &= \ddot{r} \,\hat{e}_r + \dot{r} \dot\theta \,\hat{e}_\theta + \dot{r} \dot\theta \,\hat{e}_\theta + r \ddot\theta \,\hat{e}_\theta - r \dot\theta \dot\theta \,\hat{e}_r \\ &= (\ddot{r} - r\dot\theta^2) \,\hat{e}_r + (r \ddot\theta + 2 \dot{r} \dot\theta) \,\hat{e}_\theta. \end{aligned}$

The acceleration term $-r\dot\theta^2\,\hat{e}_r$ is called the centripetal (center-seeking) acceleration, while the $2\dot{r}\dot\theta \,\hat{e}_\theta$ term is called the Coriolis acceleration.

TAM 212: Introductory Dynamics

Position, velocity, and acceleration #rkv

Position vectors #rkv‑sp

Transformation of position vectors #rkv‑st

Velocity and acceleration vectors #rkv‑sv

Velocity and acceleration in Cartesian basis #rkv‑sr

Velocity and acceleration in polar basis #rkv‑sl