28.6 Relativistic energy  (Page 7/12)

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Section summary

• Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
• Total Energy is defined as: $E={\mathrm{\gamma mc}}^2$ , where $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ .
• Rest energy is $E_0={\mathrm{mc}}^2$ , meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
• We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
• The relativistic work-energy theorem is $W_{\text{net}}=E-E_0=\gamma {\mathrm{mc}}^2-{\mathrm{mc}}^2=\left(\gamma -1\right){\mathrm{mc}}^2$ .
• Relativistically, $W_{\text{net}}={\text{KE}}_{\text{rel}}$ , where ${\text{KE}}_{\text{rel}}$ is the relativistic kinetic energy.
• Relativistic kinetic energy is ${\text{KE}}_{\text{rel}}=\left(\gamma -1\right){\mathrm{mc}}^2$ , where $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
• No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
• The equation $E^2=(\mathrm{pc}{)}^2+({\mathrm{mc}}^2{)}^2$ relates the relativistic total energy $E$ and the relativistic momentum $p$ . At extremely high velocities, the rest energy ${\mathrm{mc}}^2$ becomes negligible, and $E=\mathrm{pc}$ .

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