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 $S^{'}$ , in standard configuration
origin of $S^{'}$ moves at a speed, $\vec{u}$ , in the x direction, along the x-axis of $S$
two frames coincide at $t = 0, x = x^{'}$
at $t > 0$ : an object stationary at position $x$ in $S$ has moved a distance $- u t$ in the $x^{'}$ axis.
the galilean transformations:

(x^{'}, y^{'}, z^{'}, t^{'}) = (x - u t, y, z, t)

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

velocity addition

an object has velocity, $\vec{v} = (v_{x}^{'}, v_{y}^{'}, v_{z}^{'})$ , in frame $S^{'}$ . what is $\vec{v}$ in frame $S$ ?
- $v_{x} = \frac{d x}{d t} = \frac{d}{d t} (x^{'} + u t^{'}) = \frac{d x^{'}}{d t} + u \frac{d t^{'}}{d t} = \frac{d x^{'}}{d t^{'}} \frac{d t^{'}}{d t} + u \frac{d t^{'}}{d t}$
  $v_{x} = \frac{d x^{'}}{d t^{'}} + u$ ;
- therefore, $v_{x} = v_{x}^{'} + u$

(v_{x}, v_{y}, v_{z}) = (v_{x}^{'} + u, v_{y}^{'} + u, v_{z}^{'} + u)

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