A photon has momentum of magnitude $8.24\times {10}^{-28}$ kg-m/s. What is the energy of this photon? Give your answer in joules and in electron volts.

A particle has rest mass $6.64\times {10}^{-27}$ kg and momentum $2.10\times {10}^{-18}$ kgm/s.

(a) What is the total energy (kinetic plus rest energy) of the particle?

(b) What is the kinetic energy of the particle?

(c) What is the ratio of the kinetic energy to the rest energy of the particle?

Compute the kinetic energy of a proton (mass $1.67\times {10}^{-27}$ kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00\times {10}^7$ m/s and (b) $2.85\times {10}^8$ m/s.

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 μs. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth's surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth's surface. (a) What is the greatest distance a muon could travel during its 2.2 μs lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2 μs lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 μs, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of $0.800c$ relative to you. At the instant the spaceracer passes you, both of you start timers at zero.

(a) At the instant when you measure that the spaceracer has traveled $1.20\times {10}^8$ m past you, what does the race pilot read on her timer?

(b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her?

(c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

A photon scatters in the backward direction ($\varphi =180$°) from a free proton that is initially at rest. What must the wavelength of the incident photon be if it is to undergo a $10.0\%$ change in wavelength as a result of the scattering?

X rays with an initial wavelength of $0.900\times {10}^{-10}$ m undergo Compton scattering. For what scattering angle is the wavelength of the scattered x rays greater by $1.0\%$ than that of the incident x rays?

An electron and a positron are moving toward each other and each has speed $0.500c$ in the lab frame. What is the kinetic energy of each particle?

Model a hydrogen atom as an electron in a cubical box with side length $L$. Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a=5.29\times {10}^{-11}$ m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.