5.2 Relative Motion

Uniform Relative Motion

Recall our definition of relative position, i.e. $\overrightarrow{p}_{relative} = \overrightarrow{p}_{final} - \overrightarrow{p}_{initial}$

By using the definition from 2.1: Uniform Linear Motion, we can then define relative velocity as $\overrightarrow{v}_{relative} = \dfrac{\overrightarrow{p}_{relative}}{t}$

Rewriting with the appropriate conventional symbols, $\overrightarrow{v}_{relative} = \dfrac{\overrightarrow{s}_{relative}}{t}$

But note that this is only defining it for uniform relative motion. How do we generalize this to non-uniform relative motion?

Non-Uniform Relative Motion

This step is super easy if we use Taylor series, as discussed in 2.2: Accelerated Linear Motion and Generalization.

Hence, $s_{relative} = \sum^{m}_{n=0} \dfrac{d^n s}{dt^n} \dfrac{{x_{relative}}^n}{n!}$

Here, m is the highest degree of the rate of change. Or simply, which ultimate quantity signifying the change in position is constant.

Note that in our Taylor series, our shift of origin, ie the $x-c$ term, is within the expansion of $x_{relative}$. Note that we have not used vectors in this case for our equation. This is due to the fact that this expansion is more complex, and you can read about it here https://people.sc.fsu.edu/~jburkardt/classes/gateway_2014/lecture_week14.pdf.