28.6 Relativistic Energy

A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is $0.992 c$?

Show that relativistic kinetic energy becomes the same as classical kinetic energy when $v = c$ .

What is the rest energy of an electron, given its mass is $9 . \text{11} \times \text{10}^{- \text{31}} \text{ kg}$? Give your answer in joules and MeV.

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?

A supernova explosion of a $2 . \text{00} \times \text{10}^{\text{31}} \text{ kg}$ star produces $1 . \text{00} \times \text{10}^{\text{44}} \text{ J}$ of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio $Δ m / m$ of mass destroyed to the original mass of the star?

(a) Using data from the reference, calculate the amount of mass converted to energy by the fusion of 1.00 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, $Δ m / m$? (c) How does this compare with $Δ m / m$ for the fission of 1.00 kg of uranium?

A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron’s kinetic energy, find $\gamma$ for the electron. (b) What is the electron’s velocity?

(a) Calculate the relativistic kinetic energy of a 1000-kg car moving at 30.0 m/s if the speed of light were only 45.0 m/s. (b) Find the ratio of the relativistic kinetic energy to classical.

(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.

What is the kinetic energy in MeV of a $\pi$-meson that lives $1 . \text{40} \times 10^{- \text{16}} \text{ s}$ as measured in the laboratory, and $0 . \text{840} \times 10^{- \text{16}} \text{ s}$ when at rest relative to an observer, given that its rest energy is 135 MeV?

(a) Show that $\left(\right. pc \left.\right)^{2} / \left(\right. mc^{2} \left.\right)^{2} = \gamma^{2} - 1$. This means that at large velocities $pc >> mc^{2}$. (b) Is $E \approx \text{pc}$ when $\gamma = \text{30} . 0$, as for the astronaut discussed in the twin paradox?

What is $\gamma$ for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?

(a) Using data from the reference, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is $750 kg /\text{m}^{3}$, what is the ratio of mass destroyed to original mass, $Δ m / m$?

A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An electron?

(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is $10^{4} \text{kg}$?

The Sun produces energy at a rate of $4 . \text{00} \times \text{10}^{\text{26}}$ W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass will it have lost in…

A space rock with a length of 1,000.0 m is moving through space at exactly 0.6 c. (a) If the space rock is moving toward an observer, what is the contracted length observed? (b) If the space rock is moving away from the observer, what is the contracted length observed? (c) Can the object reach the speed of light? (d) If the object were to stop in the observer’s reference frame, would it be observ…