20.6 Relativistic energy  (Page 9/12)

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

Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900s.

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

One cosmic ray neutron has a velocity of $0.250c$ relative to the Earth. (a) What is the neutron’s total energy in MeV? (b) Find its momentum. (c) Is $E\approx \text{pc}$ in this situation? Discuss in terms of the equation given in part (a) of the previous problem.

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) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if $\gamma =1\text{.}\text{00}\times {\text{10}}^5$ for them? (b) What is their total energy (nearly the same as kinetic in this case) in GeV?

(a) Using data from [link] , 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 $\mathrm{750\; kg}{\text{/m}}^3$ , what is the ratio of mass destroyed to original mass, $\mathrm{\Delta }m/m$ ?

(a) Calculate the energy released by the destruction of 1.00 kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?

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?

Suppose you use an average of $\mathrm{500\; kW·h}$ of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the $\mathrm{500\; kW·h}$ per month rate for one year by the energy from the described mass conversion?

(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 ${10}^4\text{kg}$ ?

Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket’s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of $1.00\times {10}^5\text{kg}$ (100 tons), what total yield nuclear explosion in tons of TNT is needed?

The Sun produces energy at a rate of $4\text{.}\text{00}\times {\text{10}}^{\text{26}}$ 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 the time found in part (b)?

Unreasonable Results

A proton has a mass of $1\text{.}\text{67}\times {\text{10}}^{-\text{27}}\text{kg}$ . A physicist measures the proton’s total energy to be 50.0 MeV. (a) What is the proton’s kinetic energy? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Construct Your Own Problem

Consider a highly relativistic particle. Discuss what is meant by the term “highly relativistic.” (Note that, in part, it means that the particle cannot be massless.) Construct a problem in which you calculate the wavelength of such a particle and show that it is very nearly the same as the wavelength of a massless particle, such as a photon, with the same energy. Among the things to be considered are the rest energy of the particle (it should be a known particle) and its total energy, which should be large compared to its rest energy.

Construct Your Own Problem

Consider an astronaut traveling to another star at a relativistic velocity. Construct a problem in which you calculate the time for the trip as observed on the Earth and as observed by the astronaut. Also calculate the amount of mass that must be converted to energy to get the astronaut and ship to the velocity travelled. Among the things to be considered are the distance to the star, the velocity, and the mass of the astronaut and ship. Unless your instructor directs you otherwise, do not include any energy given to other masses, such as rocket propellants.