35. Special Relativity

The nearest star to Earth is Proxima Centauri, 4.3 light-years away. At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 5.5 years, as measured by travelers on the spacecraft?

What is the speed of a galaxy with z = 0.068?

A particle of mass m travels at a speed v = 0.32 c. At what speed will its momentum be doubled?

What is the momentum of a proton traveling at v = 0.72c?

How much work is required to accelerate a proton from rest up to a speed of 0.995c?

Estimate the speed of a galaxy, and its distance from us, if the wavelength for the hydrogen line at 434 nm is measured on Earth as being 625 nm.

What is the time for one complete revolution for a very high-energy proton in the 27-km LHC circumference?

Escape velocity from the Earth is 11.2 km/s. What would be the percent decrease in length of a 73.6-m-long spacecraft traveling at that speed as seen from Earth?

Estimate the wavelength shift for the 656.3-nm line in the Balmer series of hydrogen emitted from a galaxy whose distance from us is 6.0 x 10⁶ ly.

Two spaceships leave Earth in opposite directions, each with a speed of 0.50c with respect to Earth.

(a) What is the velocity of spaceship 1 relative to spaceship 2?

(b) What is the velocity of spaceship 2 relative to spaceship 1?

A stick of length ℓ₀, at rest in reference frame S, makes an angle θ with the x axis. In reference frame S', which moves to the right with velocity $\overrightarrow{v}$ = vî with respect to S, determine (a) the length l of the stick, and (b) the angle θ it makes with the x' axis.

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).

A particle is described by a wave function $\psi \left(x\right)=A{e}^{-\alpha x^2}$, where $A$ and $\alpha$ are real, positive constants. If the value of $\alpha$ is increased, what effect does this have on (a) the particle’s uncer­tainty in position and (b) the particle’s uncertainty in momentum? Explain your answers.

Consider a wave function given by $\psi \left(x\right)=Asinkx$, where $k=2\pi /\lambda$ and $A$ is a real constant.

(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.

(b) For which values of $x$ is the probability zero? Explain.

(a) Find the lowest energy level for a particle in a box if the particle is a billiard ball ($m=0.20$ kg) and the box has a width of $1.3$ m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls; that is, ignore the rotational kinetic energy.)

(b) Since the energy in part (a) is all kinetic, to what speed does this correspond? How much time would it take at this speed for the ball to move from one side of the table to the other?