Theory of Relativity

$$x'=\frac{x-ut}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$t'=\frac{t-x\frac{v}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$l=l_0\sqrt{1-\frac{v^2}{c^2}}$$

$$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$p=\frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$v_r=\frac{v+u}{1-\frac{vu}{c^2}}$$

$$E=E_0+E_c<=>mc^2=m_0c^2+\Delta mc^2$$

$$E=c\sqrt{p^2+m_0^2c^2}$$

$$p=\frac{1}{c}\sqrt{E_c(E_c+2E_0)}$$

Relativistic rockets

$$m=M \left [ \frac{(1+\frac{v}{c})(1-\frac{v_0}{c})}{(1-\frac{v}{c})(1+\frac{v_0}{c})} \right ]^\frac{-c}{2u}$$

Period of relativistic armonic oscillations

$$T=2\pi \sqrt{\frac{m_0}{k}}(1+\frac{3kA^2}{16m_0c^2})$$

Relativistic Doppler effect

$$\frac{\nu_1}{\nu_2}=\sqrt{\frac{1+\frac{v^2}{c^2}}{1-\frac{v^2}{c^2}}}$$