35. Special Relativity

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.

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

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.

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?

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: What is the distance traveled according to observers on the spacecraft?

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?

Starting from Eq. 36–16a, show that the Doppler shift in wavelength is, if v ≪ c, ∆λ / λ = v/c.

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. Put in values and show ∆θ = 0.87". (General Relativity predicts an angle twice as large, 1.74".)

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.]

For a 1.0-kg mass, make a plot of the kinetic energy as a function of speed for speeds from 0 to 0.9c, using both the classical formula ( K = 1/2 mv²) and the correct relativistic formula ( K = ( γ -1)mc²).

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)?

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?