35. Special Relativity

Show that the kinetic energy K of a particle of mass m is related to its momentum p by the equation $p=\frac{\sqrt{K^2+2Km{c}^2}}{c}$.

The positive muon (µ⁺), an unstable particle, lives on average 2.20 × 10^-6 s (measured in its own frame of reference) before decaying. If such a particle is moving, with respect to the laboratory, with a speed of 0.900c, what average lifetime is measured in the laboratory?

A modest supernova (the explosion of a massive star at the end of its life cycle) releases 1.5 x 10⁴⁴ J of energy in a few seconds. This is enough to outshine the entire galaxy in which it occurs. Suppose a star with the mass of our sun collides with an antimatter star of equal mass, causing complete annihilation. What is the ratio of the energy released in this star-antistar collision to the energy released in the supernova?

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.]

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

The fission process n + ²³⁵U → ²³⁶U → ¹⁴⁴Ba + ⁸⁹Kr + 3n converts 0.185 u of mass into the kinetic energy of the fission products. What is the total kinetic energy in MeV?

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?

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.

(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.]

You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. By how much? Hint: Use the binomial approximation.

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)?

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.

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?