35. Special Relativity

Carol is in the same reference frame with a clock. Bianca is flying past Carol and her clock at a high speed. Bianca sees Carol's clock ticking at one quarter the rate that Carol sees. How fast is Bianca flying relative to Carol?

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.

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?

250

Carl is standing in a park 1000 m across. Rohan flies over the park at a very high speed, first passing over the east end of the park, and then passing over the west end. Carl and Rohan are discussing the time interval between when Rohan passed over the east end of the park and when he passed over the west end of the park. Who measured the proper time?

231

According to Olive, her new super-speeder space yacht is

60 . 0 m

long. Lennon is on Earth watching Olive approaching at

0.964 c

. How long is Olive's space yacht according to Lennon?

266

As seen from a distant planet, ships A and B fly toward each other, with ship A having a speed of

0.91 c

. and ship B having a speed of

0.55 c

. According to the pilot of ship A, how fast is ship B approaching?

229

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?

(II) 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?

(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.]

A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height?

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?

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 ms. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth’s surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth’s surface. (a) What is the greatest distance a muon could travel during its 2.2@ms lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2@ms lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 ms, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon must travel, as measured by the muon? Is it now clear how the muon is able to reach the ground?

A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Tell It to the Judge. (a) How fast must you be approaching a red traffic light 1l = 675 nm2 for it to appear yellow 1l = 575 nm2? Express your answer in terms of the speed of light. (b) If you used this as a reason not to get a ticket for running a red light, how much of a fine would you get for speeding? Assume that the fine is $1.00 for each kilometer per hour that your speed exceeds the posted limit of 90 km>h

Open Question

(II) 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?

(III) 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.]

(II) Calculate the escape velocity, using Newtonian mechanics, from an object that has collapsed to its Schwarzschild radius.

A 0.30-kg meter stick moving parallel to its length passes you at high speed. You measure its length to be 48.0 cm. What is its kinetic energy?

Rocket A passes Earth at a speed of 0.65c. At the same time, rocket B passes Earth moving 0.90c relative to Earth in the same direction. How fast is B moving relative to A when it passes A?

(II) Show that the kinetic energy K of a particle of mass m is related to its momentum p by the equation

p = √ K² + (2Kmc²/c)