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S1.5 Relativity: the common sense view

[A] Context

[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 are

\[ \vec{r}_{PA} = \vec{r}_{PB} + \vec{r}_{BA} \]

\[ \vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA} \]

\[ \vec{a}_{PA} = \vec{a}_{PB} \]
These are the Galilean transformations.

They are based on the assumption that time is simple.

Graphic - No title - galileo
QIs time simple?

Analysis

Analysis

First consider 1D case:

 

Graphic - No title - galileo1D

  • 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

    $$ \vec{r}_{PA} = \vec{r}_{PB} + \vec{r}_{BA} $$

    • $\vec{r}_{PA}$ locates P w.r.t origin of A
    • $\vec{r}_{BA}$ locates origin of B w.r.t origin of A
    • note order of subscripts

    A tip!

    A tip!

    The subscripts (P and A) on the left hand side of this equation are the first and last subscripts on the right hand side. And the ‘other’ subscript on the right hand side appears twice in succession. This rule allows you to check the consistency of equations like this.
     
  • Differentiating w.r.t time gives

    \[ \frac{d\vec{r}_{PA}}{dt} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{d\vec{r}_{PB}}{dt} + \frac{d\vec{r}_{BA}}{dt}} \]

    and thence the second equation:

    $$ \vec{v}_{PA} = \vec{v}_{PB} + \vec{v}_{BA}$$

    • $\vec{v}_{PA}$ is velocity of P w.r.t A
    • $\vec{v}_{BA}$ is velocity B w.r.t A

    Differentiating w.r.t time once more gives

    \[ \frac{d\vec{v}_{PA}}{dt} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{d\vec{v}_{PB}}{dt} + \frac{d\vec{v}_{BA}}{dt}} \]

    and since $\vec{v}_{BA}$ is constant:

    $$ \vec{a}_{PA} = \vec{a}_{PB}$$

 

Commentary

Commentary

Observers using reference frames A and B assign:
  • different positions …because of different origins
  • different velocities …because of relative motion
  • same accelerations …because relative velocity is constant
 

Example

Example

Graphic - No title - galileoex
In road frame R, both G and H have speed u. Find $\vec{v}_{HG}$
  • Appeal to   $ \vec{v}_{HG}= \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\vec{v}_{HR} + \vec{v}_{RG}}$
  • Identifications:

     
    • $\vec{v}_{HR}$ is ‘west’, magnitude u
    • $\vec{v}_{RG}$ is ‘north’, magnitude u
    Graphic - No title - crasheg

  • Hence from the vector triangle:  $\vec{v}_{HG}$
    • has magnitude $\sqrt{2} u$
    • points ‘north west’
 

[C] Status of results

These results are
TVelocity of the Moon Relative to the Sun

Learning Resources

Textbook: HRW Chapter 4.8-9
Course Questions:
Self-Test Questions: