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University Physics Volume 3

# Key Equations

### Key Equations

 Time dilation $Î”t=Î”Ï„1âˆ’v2c2=Î³Ï„Î”t=Î”Ï„1âˆ’v2c2=Î³Ï„$ Lorentz factor $Î³=11âˆ’v2c2Î³=11âˆ’v2c2$ Length contraction $L=L01âˆ’v2c2=L0Î³L=L01âˆ’v2c2=L0Î³$ Galilean transformation $x=xâ€²+vt,y=yâ€²,z=zâ€²,t=tâ€²x=xâ€²+vt,y=yâ€²,z=zâ€²,t=tâ€²$ Lorentz transformation $t=tâ€²+vxâ€²/c21âˆ’v2/c2t=tâ€²+vxâ€²/c21âˆ’v2/c2$ $x=xâ€²+vtâ€²1âˆ’v2/c2x=xâ€²+vtâ€²1âˆ’v2/c2$ $y=yâ€²y=yâ€²$ $z=zâ€²z=zâ€²$ Inverse Lorentz transformation $tâ€²=tâˆ’vx/c21âˆ’v2/c2tâ€²=tâˆ’vx/c21âˆ’v2/c2$ $xâ€²=xâˆ’vt1âˆ’v2/c2xâ€²=xâˆ’vt1âˆ’v2/c2$ $yâ€²=yyâ€²=y$ $zâ€²=zzâ€²=z$ Space-time invariants $(Î”s)2=(Î”x)2+(Î”y)2+(Î”z)2âˆ’c2(Î”t)2(Î”s)2=(Î”x)2+(Î”y)2+(Î”z)2âˆ’c2(Î”t)2$ $(Î”Ï„)2=âˆ’(Î”s)2/c2=(Î”t)2âˆ’[(Î”x)2+(Î”y)2+(Î”z)2]c2(Î”Ï„)2=âˆ’(Î”s)2/c2=(Î”t)2âˆ’[(Î”x)2+(Î”y)2+(Î”z)2]c2$ Relativistic velocity addition $ux=(uxâ€²+v1+vuxâ€²/c2),uy=(uyâ€²/Î³1+vuxâ€²/c2),uz=(uzâ€²/Î³1+vuxâ€²/c2)ux=(uxâ€²+v1+vuxâ€²/c2),uy=(uyâ€²/Î³1+vuxâ€²/c2),uz=(uzâ€²/Î³1+vuxâ€²/c2)$ Relativistic Doppler effect for wavelength $Î»obs=Î»s1+vc1âˆ’vcÎ»obs=Î»s1+vc1âˆ’vc$ Relativistic Doppler effect for frequency $fobs=fs1âˆ’vc1+vcfobs=fs1âˆ’vc1+vc$ Relativistic momentum $pâ†’=Î³muâ†’=muâ†’1âˆ’u2c2pâ†’=Î³muâ†’=muâ†’1âˆ’u2c2$ Relativistic total energy $E=Î³mc2,whereÎ³=11âˆ’u2c2E=Î³mc2,whereÎ³=11âˆ’u2c2$ Relativistic kinetic energy $Krel=(Î³âˆ’1)mc2,whereÎ³=11âˆ’u2c2Krel=(Î³âˆ’1)mc2,whereÎ³=11âˆ’u2c2$
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