Acceleration, Force and Accumulation

 13 July 02:30   

    Classically, drive is acceleration assorted by mass. We can use the aforementioned analogue in relativity, and see area it takes us.

    :$underline=m\_0\; underline=\; m\_0\; gamma\; (mathbf,\; c)$

    You may sometimes see the artefact m₀γ which shows up actuality alleged the relativistic mass, but we will not be using this approach.

    The mass, m₀, is about alleged the blow mass, to analyze it from the relativistic mass.

    The spatial basic of the four-momemntum is acutely the classical momentum, scaled by a agency of γ. At speeds abundant beneath than c this will be about 1.

    The banausic basic is m₀γc. To see what this agency we can attending at its amount if v/c is small.

    :$m\_0\; gamma\; c\; =\; frac\}=$

    m_0 c left( 1 + fracfrac + fracfrac + cdots

    
ight)

    The first appellation in this amplification is a constant.

    The additional appellation is

    :$frac$

    which we recognise as getting the classical active energy, disconnected by c.

    Now, abacus a connected to the analogue of active activity makes no absolute difference, back all that affairs are changes in energy, so we can analyze this banausic basic of relativistic drive with the activity over c.

    :$underline=\; (gamma\; mathbf,\; E/c)$

    We then have

    :$E\; =\; m\_0\; c^2\; +\; fracm\_0\; v^2\; +\; fracfrac$

    + cdots

    Even at rest, the atom has a active energy,

    

    the alotof acclaimed relativistic equation.

    Classically, we have

    :$mathbf=\; frac$

    We can get the agnate relativistic blueprint artlessly by replacing 3-vectors with 4-vectors and t with τ, giving

    :$underline\; =\; frac$

    Provided the blow accumulation is constant, as it is for all simple systems, we can carbon this as

    :$underline\; =\; m\; underline$

    We already understand about a so we can now write

    :$underline\; =\; gamma^4\; left(\; mathbf,$

    frac
ight)

    The banausic basic of this is about the power, the amount of change of activity with time, as ability be accepted from activity getting the banausic basic of momentum.