A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 32 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy.
(a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed.
(b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

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?

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).

Using Example 36–2 as a guide, show that for objects that move slowly in comparison to c, the length contraction formula is roughly ℓ ≈ ℓ₀ (1 - 1/2 v²/c²) . Use this approximation to find the “length shortening” ∆ℓ = ℓ₀ - ℓ of the train in Example 36–6 if the train travels at 100 km/h (rather than 0.92c).

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 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).

Two protons, each having a speed of 0.945c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton.