S1.5 Relativity: the common sense view
[A] Context
The description given to the motion of any object depends upon the perspective of the observer.
- A reference frame is the name given to the coordinate system to which an observer (or group of observers) refers measurements.
- Relativity is concerned with the relationship between measurements referred to different reference frames.
[B] Results: Galilean transformations
Key Point 1.9
The relationships between the positions, velocities and accelerations of a particle, P, assigned in two reference frames, A and B, in uniform relative motion areThey are based on the assumption that time is simple. |
Analysis
First consider 1D case:
- Alison stands by road (frame A)
- Billy passes along in car (frame B)
- Plane passes overhead
- Then xPA=xPB+xBA
- Here xPA is position of P w.r.t. A (1D vector from A to P)
- Differentiating w.r.t. time:
vPA=vPB+vBA
- Here vPA is velocity of P w.r.t. A
Now consider general (2D) case:
Inspection of figure gives first equation

locates P w.r.t origin of A
locates origin of B w.r.t origin of A- note order of subscripts
Differentiating w.r.t time gives
![\[ \frac{d\vec{r}_{PA}}{dt} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{d\vec{r}_{PB}}{dt} + \frac{d\vec{r}_{BA}}{dt}} \]](mastermathpng-6.png)
and thence the second equation:

is velocity of P w.r.t A
is velocity B w.r.t A
Differentiating w.r.t time once more gives
![\[ \frac{d\vec{v}_{PA}}{dt} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{d\vec{v}_{PB}}{dt} + \frac{d\vec{v}_{BA}}{dt}} \]](mastermathpng-10.png)
and since
is constant:
[C] Status of results
These results are- consistent with common sense
- practically correct for ‘slow’ kinematics
- wrong for ‘fast’ kinematics, where ‘fast’ signifies involvement of speeds comparable with speed of light, c.
Learning Resources
| HRW Chapter 4.8-9 | |



