5.9 Relativistic Energy

What is the kinetic energy of an electron if its speed is 0.992c?

What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.

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.

Given the fact that light travels at c , can it have mass? Explain.

What is the rest energy of an electron, given its mass is $9.11 \times 10^{- 31} \text{kg} ?$ 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 mass in kilograms?

A supernova explosion of a $2.00 \times 10^{31} \text{kg}$ star produces $1.00 \times 10^{44} \text{J}$ of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio $\Delta m / m$ of mass destroyed to the original mass of the star?

(a) Using data from this table, 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, $\Delta m / m$ ? (c) How does this compare with $\Delta m / m$ 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.

(a) At what relative velocity is $\gamma = 2.00 ?$ (b) At what relative velocity is $\gamma = 10.0 ?$

(a) How long does it take the astronaut in the reference to travel 4.30 ly at $0.99944 c$ (as measured by the earthbound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\gamma = 30.00$ as given.

(a) Find the value of $\gamma$ for the following situation. An astronaut measures the length of their spaceship to be 100 m, while an earthbound observer measures it to be 25.0 m. (b) What is the speed of the spaceship relative to Earth?

An astronaut has a heartbeat rate of 66 beats per minute as measured during his physical exam on Earth. The heartbeat rate of the astronaut is measured when he is in a spaceship traveling at 0.5c with respect to Earth by an observer (A) in the ship and by an observer (B) on Earth. (a) Describe an experimental method by which observer B on Earth will be able to determine the heartbeat rate of the a…

Same two observers as in the preceding exercise, but now we look at two events occurring in spaceship A. A photon arrives at the origin of A at its time $t = 0$ and another photon arrives at $\left(x = 1.00 \text{m} , 0 , 0\right)$ at $t = 0$ in the frame of ship A. (a) Find the coordinates and times of the two events as seen by an observer in frame B. (b) In which frame are the two events simultaneous and in which frame are they are not simultane…

An observer at origin of inertial frame S sees a flashbulb go off at $x = 150 \text{km} , y = 15.0 \text{km} ,$ and $z = 1.00 \text{km}$ at time $t = 4.5 \times 10^{−4} \text{s} .$ At what time and position in the S$'$ system did the flash occur, if S$'$ is moving along shared x-direction with S at a velocity $v = 0.6 c ?$

An observer standing by the railroad tracks sees two bolts of lightning strike the ends of a 500-m-long train simultaneously at the instant the middle of the train passes him at 50 m/s. Use the Lorentz transformation to find the time between the lightning strikes as measured by a passenger seated in the middle of the train.

Two astronomical events are observed from Earth to occur at a time of 0.30 s apart and a distance separation of $2.0 \times 10^{9} \text{m}$ from each other. How fast must a spacecraft travel from the site of one event toward the other to make the events occur at the same time when measured in the frame of reference of the spacecraft?

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $12.0 \times 10^{9} \text{ly}$ away is receding from us at 0.900c, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at 0.990c as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from Earth? You may assume that the velocity of the other galaxy…

Repeat the preceding problem with the ship heading directly away from Earth.

(a) Suppose the speed of light were only 3000 m/s. A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1000 m/s. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

A space probe speeding towards the nearest star moves at $0.250 c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on Earth?

(a) Calculate the speed of a $1.00 - \mu\text{g}$ particle of dust that has the same momentum as a proton moving at 0.999c. (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

Show that the relativistic form of Newton’s second law is (a) $F = m \frac{d u}{d t} \frac{1}{\left(1 - u^{2} / c^{2}\right)^{3 / 2}} ;$ (b) Find the force needed to accelerate a mass of 1 kg by 1 $\text{m}/\text{s}^{2}$ when it is traveling at a velocity of c/2.

What is the kinetic energy in MeV of a π-meson that lives $1.40 \times 1 0^{−16} \text{s}$ as measured in the laboratory, and $0.840 \times 1 0^{−16} \text{s}$ when at rest relative to an observer, given that its rest energy is 135 MeV?

(a) Show that $\left(p c\right)^{2} / \left(m c^{2}\right)^{2} = \gamma^{2} - 1 .$ This means that at large velocities $p c > > m c^{2} .$ (b) Is $E \approx p c$ when $\gamma = 30.0 ,$ 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)?

(a) Using data from Potential Energy of a System, 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 $750 \text{kg}/\text{m}^{3} ,$ what is the ratio of mass destroyed to original mass, $\Delta m / m ?$

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 $3.85 \times 10^{26}$ W by the fusion of hydrogen. About 0.7% of each kilogram of hydrogen goes into the energy generated by the Sun. (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 mas…