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
$E=\mathrm{pc}$ .
Conceptual questions
How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?
Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.
The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.
If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon’s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)
What is the rest energy of an electron, given its mass is
$9\text{.}\text{11}\times {\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ ? Give your answer in joules and MeV.
Find the rest energy in joules and MeV of a proton, given its mass is
$1\text{.}\text{67}\times {\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .
If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?
Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.077 m2, and the magnitude of the fluid velocity is 3.50 m/s.
(a) What is the fluid speed at points in the pipe where the cross
the branch of science concerned with the nature and properties of matter and energy. The subject matter of physics includes mechanics, heat, light and other radiation, sound, electricity, magnetism, and the structure of atoms.
Aluko
and the word of matter is anything that have mass and occupied space