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Relativistic Mechanics

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  • Lorentz contraction of length and slowing of a moving clock: $$l = l_0 \sqrt{1 - (v/c)^2}, \; \Delta t = \frac{\Delta t_0}{\sqrt{1 - (v/c)^2}},$$ where $l_0$ is the proper length and $\Delta t_0$ is the proper time of the moving clock.
  • Lorentz transformation*: $$ x^\prime = \frac{x - Vt}{\sqrt{1 - (V/c)^2}}, \; y^\prime = y, \; t^\prime = \frac{t-x V/c^2}{1-(V/c)^2}.$$
  • Interval $s_{12}$ is an invariant: $$s^2_{12} = c^2 t^2_{12} - l^2_{12} = inv,$$ where $t_{12}$ is the time interval between events 1 and 2, $l_{12}$ is the distance between the points at which these events occurred.
  • Transformation of velocity*: $$v_x^\prime = \frac{v_x - V}{1-v_x V/c^2}, \; v_y^\prime = \frac{v_y \sqrt{1 - (V/c)^2}}{1 - v_x V/c^2}.$$
  • Relativistic mass and relativistic momentum: $$m = \frac{m_0}{\sqrt{1-(v/c)^2}}, \; \vec{p} = m \vec{v} = \frac{m_0 \vec{v}}{\sqrt{1-(v/c)^2}},$$ where $m_0$ is the rest mass, or, simply, the mass.
  • Relativistic equation of dynamics for a particle: $$\frac{d\vec{p}}{dt} = \vec{F},$$ where $\vec{p}$ is the relativistic momentum of the particle.
  • Total and kinetic energies of a relativistic particle: $$E = mc^2 = m_0 c^2 + T, \; T = (m-m_0)c^2.$$
  • Relationship between the energy and momentum of a relativistic particle $$E^2 - p^2c^2 = m_0^2 c^4, \; pc = \sqrt{T(T+2m_0 c^2)}.$$
  • When considering the collisions of particles it helps to use the following invariant quantity: $$E^2 - p^2c^2 = m_0^2 c^4,$$ where $E$ and $p$ are the total energy and momentum of the system prior to the collision, and $m_0$ is the rest mass of the particle (or the system) formed.


* The reference frame $!K^\prime$ is assumed to move with a velocity $V$ in the positive direction of the $x$ axis of the frame $K$, with the $!x^\prime$ and $x$ axes coinciding and the $!y^\prime$ and $y$ axes parallel.

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