35. Special Relativity

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?

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overrightarrow{L}$ and the $z$-axis is $u=26.6$°. If this is the smallest angle for this particular value of the orbital quantum number $l$, what is $l$?

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.

A quarter-pound hamburger with all the fixings has a mass of 200 g. The food energy of the hamburger (480 food calories) is 2 MJ. By what factor does the energy equivalent exceed the food energy?

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?

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

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.

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?

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.

Some particle accelerators allow protons (p⁺) and antiprotons (p⁻) to circulate at equal speeds in opposite directions in a device called a storage ring. The particle beams cross each other at various points to cause p⁺ + p⁻ collisions. In one collision, the outcome is p⁺ + p⁻ → e⁺ + e⁻ + γ + γ, where γ represents a high-energy gamma-ray photon. The electron and positron are ejected from the collision at 0.9999995c and the gamma-ray photon wavelengths are found to be 1.0 x 10^-6 nm. What were the proton and antiproton speeds, as a fraction of c, prior to the collision?

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?

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?

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?

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).

Which of the following is a key difference between $Newton’s$ laws of motion and $Einstein’s$ theory of relativity?