Suppose the two lightning bolts shown in Fig. 37.5a are simultaneous to an observer on the train. Show that they are not simultaneous to an observer on the ground. Which lightning strike does the ground observer measure to come first?

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?

The positive muon (µ⁺), an unstable particle, lives on average 2.20 × 10^-6 s (measured in its own frame of reference) before decaying. What average distance, measured in the laboratory, does the particle move before decaying?

As you pilot your space utility vehicle at a constant speed toward the moon, a race pilot flies past you in her spaceracer at a constant speed of $0.800c$ relative to you. At the instant the spaceracer passes you, both of you start timers at zero.

(a) At the instant when you measure that the spaceracer has traveled $1.20\times {10}^8$ m past you, what does the race pilot read on her timer?

(b) When the race pilot reads the value calculated in part (a) on her timer, what does she measure to be your distance from her?

(c) At the instant when the race pilot reads the value calculated in part (a) on her timer, what do you read on yours?

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for $0.150$ s. The first officer on the spacecraft measures that the searchlight is on for $12.0$ ms.

(a) Which of these two measured times is the proper time?

(b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light $c$?

Why Are We Bombarded by Muons? Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 μs. 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 μs 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 μs 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 μs, 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?

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?

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?

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600c. The pursuit ship is traveling at a speed of 0.800c relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured in the laboratory?

Tell It to the Judge. (a) How fast must you be approaching a red traffic light (λ = 675 nm) for it to appear yellow (λ = 575 nm)? 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.

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?

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s² in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

A proton has momentum with magnitude p₀ when its speed is 0.400c. In terms of p₀, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

A proton (rest mass $1.67\times {10}^{-27}$ kg) has total energy that is $4.00$ times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Electrons are accelerated through a potential difference of $750$ kV, so that their kinetic energy is $7.50\times {10}^5$ eV.

(a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c$?

(b) What would the speed be if it were computed from the principles of classical mechanics?

An observer in frame S′ is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in S′ measures the speed v′ of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a) v′ = 0.400c; (b) v′ = 0.900c; (c) v′ = 0.990c?

A particle has rest mass $6.64\times {10}^{-27}$ kg and momentum $2.10\times {10}^{-18}$ kgm/s.

(a) What is the total energy (kinetic plus rest energy) of the particle?

(b) What is the kinetic energy of the particle?

(c) What is the ratio of the kinetic energy to the rest energy of the particle?

Compute the kinetic energy of a proton (mass $1.67\times {10}^{-27}$ kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00\times {10}^7$ m/s and (b) $2.85\times {10}^8$ m/s.