Relative Motion, Velocity, and Acceleration

Last edited: 2024-11-07 13:18:16

Two moving objects' motion relative to each other dictates the dynamics of their relative position, velocity, and acceleration.

Relative Position Formula

The equation for the relative position between an object $A$ and an object $B$ looks like this:

$$\vec{r}_A = \vec{r}_B + \vec{r}_{A/B},$$

where $\vec{r}_A$ and $\vec{r}_B$ are the absolute positions of the two objects in a fixed coordinate system ($XY$ coordinate system in the figure above). $\vec{r}_{A/B}$ is the position vector of object $A$ relative to object $B$, "$A/B$" means "$A$ relative to $B$". Another way of thinking about it is that $\vec{r}_{A/B}$ is object $A$'s coordinates in a coordinate system whose origin follows object $B$ ($xy$ coordinate system in the figure above).

Relative Velocity Formula

If we differentiate the equation for the relative position we get the velocities:

$$\frac{d\vec{r}_A}{dt} = \frac{d\vec{r}_B}{dt} + \frac{d\vec{r}_{A/B}}{dt},$$

which can be rewritten as

$$\vec{v}_A = \vec{v}_B + \vec{v}_{A/B},$$

where $\vec{v}_A$ and $\vec{v}_B$ are the absolute velocities of the two objects in a fixed coordinate system and $\vec{v}_{A/B}$ is the velocity of object $A$ relative to object $B$.

Relative Velocity Due to Rotation

Relative velocity can also be used between two points on a rotating rigid body, see the figure above. This means that the distance ($\vec{r}_{A/B}$) between the two points remains constant. Thus the relative velocity vector $\vec{v}_{A/B}$ depends on the angular velocity $\omega$ around the origin of the translating coordinate system attached to point $B$ according to the following formula:

$$v_{A/B} = \omega r_{A/B}.$$

This can be written in vector form with this:

$$\vec{v}_{A/B} = \vec{\omega} \times \vec{r}_{A/B},$$

where $\vec{\omega}$ is the angular velocity vector normal to the plane of motion. Therefore the relative velocity is always perpendicular to $\vec{r}_{A/B}$. So the formula for the velocity of point $A$ is:

$$\vec{v}_A = \vec{v}_B + \vec{\omega} \times \vec{r}_{A/B}.$$

Relative Acceleration Formula

If we differentiate the equation for relative position a second time we get the accelerations:

$$\frac{d^2\vec{r}_A}{dt^2} = \frac{d^2\vec{r}_B}{dt^2} + \frac{d^2\vec{r}_{A/B}}{dt^2},$$

which can be rewritten as

$$\vec{a}_A = \vec{a}_B + \vec{a}_{A/B},$$

where $\vec{a}_A$ and $\vec{a}_B$ are the absolute accelerations of the two objects in a fixed coordinate system and $\vec{a}_{A/B}$ is the acceleration of object $A$ relative to object $B$. We can note here that if object $B$ is moving at a constant velocity, the acceleration of object $A$ will be the same in both the fixed coordinate system and the translating coordinate system attached to object $B$.

Relative Acceleration Due to Rotation

As mentioned for the relative velocity due to rotation the distance ($\vec{r}_{A/B}$) between the two points remains constant, as can be seen in the figure above. This means that the observer moving with point $B$ perceives point $A$ to have circular motion around point $B$. This means the relative acceleration will have a normal component parallel with $\vec{r}_{A/B}$ from $A$ pointing to $B$. This is because of the change in direction of $\vec{r}_{A/B}$, like centripetal acceleration, so it has the following expression:

$$(a_{A/B})_n = \frac{v_{A/B}^2}{r_{A/B}} = \omega^2 r_{A/B}$$

It will also have a tangential component due to the change in magnitude of $\vec{r}_{A/B}$ and it looks like this:

$$(a_{A/B})_t = \frac{dv_{A/B}}{dt} = \alpha r_{A/B},$$

where $\alpha$ is the angular acceleration. In vector form both equations look like this:

$$(\vec{a}_{A/B})_n = \vec{\omega} \times (\vec{\omega} \times \vec{r}_{A/B})$$ $$(\vec{a}_{A/B})_t = \vec{\alpha} \times \vec{r}_{A/B}.$$

So the formula for the relative acceleration due to rotation looks like this:

$$\vec{a}_A = \vec{a}_B + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{A/B}) + \vec{\alpha} \times \vec{r}_{A/B}.$$