35. Special Relativity

Carol is in the same reference frame with a clock. Bianca is flying past Carol and her clock at a high speed. Bianca sees Carol's clock ticking at one quarter the rate that Carol sees. How fast is Bianca flying relative to Carol?

The international space station travels in orbit at a speed of 7.67 km/s. If an astronaut and his brother start a stop watch at the same time, on Earth, and then the astronaut spends 6 months on the space station, what is the difference in time on their stopwatches when the astronaut returns to Earth? Note that 6 months is about 1.577 x 10⁷ s, and c = 3 x 10 ⁸ m/s.

In the following figure, a right triangle is shown in its rest frame, S'. In the lab frame, S, the triangle moves with a speed v. How fast must the triangle move in the lab frame so that it becomes an isosceles triangle?

256

Carl is standing in a park 1000 m across. Rohan flies over the park at a very high speed, first passing over the east end of the park, and then passing over the west end. Carl and Rohan are discussing the time interval between when Rohan passed over the east end of the park and when he passed over the west end of the park. Who measured the proper time?

236

According to Olive, her new super-speeder space yacht is

60 . 0 m

long. Lennon is on Earth watching Olive approaching at

0.964 c

. How long is Olive's space yacht according to Lennon?

273

As seen from a distant planet, ships A and B fly toward each other, with ship A having a speed of

0.91 c

. and ship B having a speed of

0.55 c

. According to the pilot of ship A, how fast is ship B approaching?

234

Bjorn is standing at x = 600 m. Firecracker 1 explodes at the origin and firecracker 2 explodes at x = 900 m. The flashes from both explosions reach Bjorn's eye at t = 3.0 μs. At what time did each firecracker explode?

(II) 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 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?

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 ms. 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@ms 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@ms 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 ms, 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?

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 1l = 675 nm2 for it to appear yellow 1l = 575 nm2? 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

(II) 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: (b) on the spacecraft?

(II) 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: (a) 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.]

(II) 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.

(II) A fictional news report stated that starship Enterprise had just returned from a 5-year voyage while traveling at 0.80c. (a) If the report meant 5.0 years of Earth time, how much time elapsed on the ship? (b) If the report meant 5.0 years of ship time, how much time passed on Earth?

(II) What is the speed of a pion if its average lifetime is measured to be 4.80 x 10⁻⁸ s? At rest, its average lifetime is 2.60 x 10⁻⁸ s.

(II) A spaceship traveling at 0.76c away from Earth fires a module with a speed of 0.85c at right angles to its own direction of travel (as seen by the spaceship). What is the speed of the module, and its direction of travel (relative to the spaceship’s direction), seen by an observer on Earth?

(II) 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: (c) What is the distance traveled according to observers on the spacecraft? (d) What will the spacecraft occupants compute their speed to be from the results of (b) and (c)?

(II) At what speed v will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)?

(II) You travel to a star 115 light-years from Earth at a speed of 2.90 x 10⁸ m/s. What do you measure this distance to be?

A healthy astronaut’s heart rate is 60 beats/min . Flight doctors on Earth can monitor an astronaut’s vital signs remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her heart rate as 52 beats/min?

(II) Starting from Eq. 36–16a, show that the Doppler shift in wavelength is, if v ≪ c ,

∆λ / λ = v/c .

(a) Use special relativity and Newton’s law of gravitation to show that a photon of mass m = E/c² just grazing the Sun will be deflected by an angle ∆θ given by

∆θ = 2GM/c²R

where G is the gravitational constant, R and M are the radius and mass of the Sun, and c is the speed of light. (b) Put in values and show ∆θ = 0.87" . (General Relativity predicts an angle twice as large, 1.74" .)

(III) Calculate the maximum kinetic energy of the electron when a muon decays from rest via μ⁻ ⟶ e⁻ + vₑ + vμ. [Hint: In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.]

(II) At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons (M ≈ 500 MeV/ c²) , (b) Y (M ≈ 9500 MeV/c²), and (c) muons ( M ≈ 100 MeV/c²)?

(II) Calculate the peak wavelength of the CMB at 1.0 s after the birth of the universe. In what part of the EM spectrum is this radiation?

(III) Starting from Eq. 44–3, show that the Doppler shift in wavelength is ∆λ/λᵣₑₛₜ ≈ v/c (Eq. 44–6) for v ≪ c . [Hint: Use the binomial expansion.]

(I) If a galaxy is traveling away from us at 2.2% of the speed of light, roughly how far away is it?

(II) Calculate the escape velocity, using Newtonian mechanics, from an object that has collapsed to its Schwarzschild radius.

Astronomers measure the distance to a particular star to be 6.0 light-years (1ly = distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.25 years as measured by clocks on the spaceship. (a) How long does the trip take as measured by clocks in Earth’s reference frame? (b) What distance does the spaceship travel as measured in its own reference frame?

A 0.30-kg meter stick moving parallel to its length passes you at high speed. You measure its length to be 48.0 cm. What is its kinetic energy?

Astronomers have measured the rotation of gas around a possible supermassive black hole of about 2 billion solar masses at the center of a galaxy. If the radius from the galactic center to the gas clouds is 68 light-years, estimate the value of z.

Rocket A passes Earth at a speed of 0.65c. At the same time, rocket B passes Earth moving 0.90c relative to Earth in the same direction. How fast is B moving relative to A when it passes A?

The Sun radiates energy at a rate of about 4 x 10²⁶ W . (a) At what rate is the Sun’s mass decreasing? (b) How long does it take for the Sun to lose a mass equal to that of Earth? (c) Estimate how long the Sun could last if it radiated constantly at this rate.

If E is the total energy of a particle with zero potential energy, show that dE/dp = v, where p and v are the momentum and velocity of the particle, respectively.

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 37–5.)

An electron (m = 9.11 x 10⁻³¹ kg) is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by 7.20 x 10⁻¹⁴ J . Determine the electron’s speed, v.

(II) Show that the kinetic energy K of a particle of mass m is related to its momentum p by the equation

p = √ K² + (2Kmc²/c)

(II) Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

A pi meson of mass m_π decays at rest into a muon (mass m_μ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is K_μ = (m_π - m_μ)² c² / (2m_π) .