2.5 Transforming Between Reference Frames

To transform between one frame of reference to another, we subtract the constant velocity between the frames from the position vector.

\boldsymbol{r}^{\prime}(t)=\boldsymbol{r}(t)-Vt.

(2.42)

We can then transform the velocity and acceleration as

	$\displaystyle\boldsymbol{v}^{\prime}(t)$	$\displaystyle=\frac{\operatorname{\mathrm{d}}\!\boldsymbol{r}^{\prime}(t)}{% \operatorname{\mathrm{d}}\!t}=\frac{\operatorname{\mathrm{d}}\!\boldsymbol{r}(% t)}{\operatorname{\mathrm{d}}\!t}-V=\boldsymbol{v}(t)-V$		(2.43)
	$\displaystyle\boldsymbol{a}^{\prime}(t)$	$\displaystyle=\frac{\operatorname{\mathrm{d}}\!^{2}\boldsymbol{r}^{\prime}(t)}% {\operatorname{\mathrm{d}}\!t^{2}}=\frac{\operatorname{\mathrm{d}}\!^{2}% \boldsymbol{r}(t)}{\operatorname{\mathrm{d}}\!t^{2}}=\boldsymbol{a}(t).$		(2.44)

One of the most useful reference frames to transform into is the centre of mass frame.