University Physics Volume 3

# Key Equations

### Key Equations

 Time dilation $Δt=Δτ1−v2c2=γτΔt=Δτ1−v2c2=γτ$ Lorentz factor $γ=11−v2c2γ=11−v2c2$ Length contraction $L=L01−v2c2=L0γL=L01−v2c2=L0γ$ Galilean transformation $x=x′+vt,y=y′,z=z′,t=t′x=x′+vt,y=y′,z=z′,t=t′$ Lorentz transformation $t=t′+vx′/c21−v2/c2t=t′+vx′/c21−v2/c2$ $x=x′+vt′1−v2/c2x=x′+vt′1−v2/c2$ $y=y′y=y′$ $z=z′z=z′$ Inverse Lorentz transformation $t′=t−vx/c21−v2/c2t′=t−vx/c21−v2/c2$ $x′=x−vt1−v2/c2x′=x−vt1−v2/c2$ $y′=yy′=y$ $z′=zz′=z$ Space-time invariants $(Δs)2=(Δx)2+(Δy)2+(Δz)2−c2(Δt)2(Δs)2=(Δx)2+(Δy)2+(Δz)2−c2(Δt)2$ $(Δτ)2=−(Δs)2/c2=(Δt)2−[(Δx)2+(Δy)2+(Δz)2]c2(Δτ)2=−(Δs)2/c2=(Δt)2−[(Δx)2+(Δy)2+(Δz)2]c2$ Relativistic velocity addition $ux=(ux′+v1+vux′/c2),uy=(uy′/γ1+vux′/c2),uz=(uz′/γ1+vux′/c2)ux=(ux′+v1+vux′/c2),uy=(uy′/γ1+vux′/c2),uz=(uz′/γ1+vux′/c2)$ Relativistic Doppler effect for wavelength $λobs=λs1+vc1−vcλobs=λs1+vc1−vc$ Relativistic Doppler effect for frequency $fobs=fs1−vc1+vcfobs=fs1−vc1+vc$ Relativistic momentum $p→=γmu→=mu→1−u2cp→=γmu→=mu→1−u2c$ Relativistic total energy $E=γmc2,whereγ=11−u2c2E=γmc2,whereγ=11−u2c2$ Relativistic kinetic energy $Krel=(γ−1)mc2,whereγ=11−u2c2Krel=(γ−1)mc2,whereγ=11−u2c2$