35. Special Relativity

What is the velocity, as a fraction of c, of an electron with 2.0 GeV total energy? Hint: This problem uses relativity.

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overrightarrow{L}$ and the $z$-axis is $u=26.6$°. If this is the smallest angle for this particular value of the orbital quantum number $l$, what is $l$?

Two identical black holes form a binary system and are orbiting one another. Assume they are a distance apart which is twice the Schwartzchild radius in each. Then, assuming Newton mechanics is still valid, how fast are they moving with respect to the center of mass?

(III) A certain atom emits light of frequency ƒ₀ when at rest. A monatomic gas composed of these atoms is at temperature T. Some of the gas atoms move toward, and others away from, an observer due to their random thermal motion. Using the rms speed of thermal motion, (a) show that the fractional difference between the Doppler-shifted frequencies for atoms moving directly toward the observer and directly away from the observer is ∆ƒ/ƒ₀ ≈ 2 √3kT/mc². Assume mc² ≫ 3kT. (b) Evaluate ∆ƒ/ƒ₀ for a gas of hydrogen atoms at 650 K. [This “Doppler-broadening” effect is commonly used to measure gas temperature, such as in astronomy.]

A quasar emits familiar hydrogen lines whose wavelengths are 8.5% longer than what we measure in the laboratory.

(a) Using the Doppler formula for light, estimate the speed of this quasar.

(b) What result would you obtain if you used the “classical” Doppler shift discussed in Chapter 16?

A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height?

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth’s surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle’s frame. (c) In the particle’s frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Tell It to the Judge. (a) How fast must you be approaching a red traffic light (λ = 675 nm) for it to appear yellow (λ = 575 nm)? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km/h.

A distant quasar is found to be moving away from the earth at 0.80c. A galaxy closer to the earth and along the same line of sight is moving away from us at 0.20c. What is the recessional speed of the quasar, as a fraction of c, as measured by astronomers in the other galaxy?

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers on the spacecraft?

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers on Earth?

Protons from outer space crash into the Earth’s atmosphere at a high rate and create particles that eventually decay into other particles called muons. These “cosmic rays” travel through the atmosphere. Every second, dozens of muons pass through your body. If a muon is created 30 km above the Earth’s surface, what minimum speed and kinetic energy must the muon have in order to hit Earth’s surface? A muon’s mean lifetime (at rest) is 2.20μs and its mass is 105.7 MeV/c².

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 1.5 years has elapsed? [Hint: Use the binomial expansion, Appendix A–2.]

How fast must a pion be moving on average to travel 28 m before it decays? The average lifetime, at rest, is 2.6 x 10^-8 s.