PX155 - G1 - gallilean transforms

statement

• "the laws of mechanics are the same in all uniformly moving frames of reference."
• all velocity is relative, there is no absolute velocity, and no universal definition of 'at rest', it depends on frames of reference

galliean transformations (GTs)

• two frames ,$S$ and $S^{\prime }$, in standard configuration
  
• origin of $S^{\prime }$ moves at a speed, $\overrightarrow{u}$, in the x direction, along the x-axis of $S$
• two frames coincide at $t=0,x=x^{\prime }$
• at $t>0$: an object stationary at position $x$ in $S$ has moved a distance $-ut$ in the $x^{\prime }$ axis.
• the galilean transformations:

• event: happens at a certain position at a certain time, $(x,y,z,t)$

velocity addition

• an object has velocity, $\overrightarrow{v}=(v_x^{\prime },v_y^{\prime },v_z^{\prime })$, in frame $S^{\prime }$. what is $\overrightarrow{v}$ in frame $S$?
  • $v_x=\frac{dx}{dt}=\frac{d}{dt}(x^{\prime }+u{t}^{\prime })=\frac{d{x}^{\prime }}{dt}+u\frac{d{t}^{\prime }}{dt}=\frac{d{x}^{\prime }}{d{t}^{\prime }}\frac{d{t}^{\prime }}{dt}+u\frac{d{t}^{\prime }}{dt}$
    $v_x=\frac{d{x}^{\prime }}{d{t}^{\prime }}+u$;
  • therefore, $v_x=v_x^{\prime }+u$

• same method is applied but different results are obtained in special relativity