Chapter 13: Special Relativity — Review and Exercises

A passenger in a smoothly cruising airplane treats the airplane as stationary and the ground as moving. Is this a valid approximately inertial frame? Must the passenger and a ground observer assign the same velocity and kinetic energy to a suitcase resting in the cabin? Explain why different measurements do not imply different laws.

An airplane moves at 300 m/s relative to the ground. The cabin air is at rest relative to the airplane, and sound travels at 340 m/s relative to that air. Using classical velocity addition, find the ground-frame velocities of sound traveling toward the front and toward the back. Take the airplane’s motion as positive. What speed does a passenger measure for either sound wave? Why is sound different from light in vacuum?

Michelson and Morley expected motion through a stationary luminiferous aether to produce a change in their interference pattern when the apparatus was rotated. Why was the absence of the expected change scientifically important? Find another scientifically significant experiment with a null result, in physics or another science. State what effect was sought, what was observed, and why the result mattered. Cite your source.

State Einstein’s two postulates of special relativity. Which one conflicts with classical velocity addition, and how?

Maxwell’s equations predict a definite speed $c$ for electromagnetic waves in vacuum. If the laws of electromagnetism are the same in every inertial frame, why would assigning light a speed $c+v$ in a frame moving relative to its source create a problem? Why does the varying measured speed of sound not create the same problem?

A spacecraft moves at $0.500c$ relative to Earth and emits light forward and backward through vacuum. What speed does each beam have in the spacecraft frame? In the Earth frame? Explain using a postulate rather than a velocity-addition calculation.

Two events at different locations are simultaneous in one inertial frame. Must they be simultaneous in every other inertial frame? Does relativity of simultaneity mean that time is merely subjective, or that no two events can ever be simultaneous? Explain how the relativistic view differs from classical absolute time.

Light from two distant events reaches you at the same instant. Is that alone enough to conclude that the events occurred simultaneously in your frame? Explain what else you need to know. After correcting for light-travel time, must all inertial frames agree that two spatially separated events were simultaneous?

A light pulse is emitted from the midpoint of a train moving to the right. In the train frame, the pulse reaches detectors at the front and rear simultaneously. In the station frame, which detector does it reach first? Explain using the motion of the detectors and the constant speed of light. Are the two detection events simultaneous in both frames?

Two light pulses reach the same detector at the same place and time. Can an observer moving relative to the detector say that one pulse reached it before the other? Explain why this differs from asking whether events at two separated detectors are simultaneous.

On a space-time diagram with position $x$ on the horizontal axis and $ct$ on the vertical axis and the same length scale on both axes, sketch worldlines starting at the origin for (a) an object at rest, (b) an object moving steadily to the right at $0.500c$, and (c) a light pulse moving right in vacuum. Which line is vertical? Which makes a 45° angle with the horizontal axis? What feature of the diagram represents two events simultaneous in this frame?

A light clock consists of two mirrors and a light pulse that travels back and forth between them. The clock moves parallel to the laboratory floor, with its mirrors separated vertically. Relative to the laboratory, the pulse follows a longer diagonal path than in the clock’s rest frame. Without doing algebra, explain why the laboratory measures more time between ticks.

A clock travels at constant velocity relative to a laboratory. Does it run abnormally according to someone traveling with it? For two ticks of that clock, which frame measures the proper time? Which measures the longer interval?

For speed $v$, define $\beta =\frac{v}{c}$ and $\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$. (a) What ranges of $\beta$ and $\gamma$ are possible for an object with nonzero rest mass? (b) What do $\beta =0$ and $\gamma$ very close to 1 tell you? (c) Can $\gamma$ exceed 1 even though $\beta$ cannot?

Calculate $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ for $v=0.001c$, $0.500c$, and $0.99999999c$. Keep enough digits in the first answer to show its difference from 1. What does the pattern tell you about ordinary speeds and speeds close to $c$?

Calculate $\gamma$ for $\beta =0.9999$ and $\beta =0.99999$, keeping all supplied digits. Approximately how many times larger is the second $\gamma$? Both speeds might round to $c$ at low precision. Why would using that rounded value in the formula for $\gamma$ be a serious error?

Use $\frac{v}{c}=\sqrt{1-\frac{1}{{\gamma }^2}}$ to find the speeds corresponding to $\gamma =2.00$ and $\gamma =12.0$. Give enough digits to distinguish each speed from $c$. Can increasing $\gamma$ make the speed of a massive object exceed $c$?

An object has $\gamma =1+5\times {10}^{\mathrm{-15}}$. Use the low-speed approximation $\gamma -1\approx \frac{{\beta }^2}{2}$ to estimate $\beta$ and $v$, taking $c=3.00\times {10}^8\text{m/s}$. Compare the speed with everyday travel. Why would rounding $\gamma$ to 1 before calculating give a misleading answer?

A hypothetical unstable particle has a mean lifetime of 2.00 μs in its rest frame. Predict whether its laboratory mean lifetime is longer or shorter, then calculate it for speeds $0.500c$, $0.900c$, and $0.9995c$.

A hypothetical unstable particle has a mean lifetime of 0.840 μs in its rest frame and 1.40 μs in the laboratory. Find $\gamma$ and its laboratory speed as a fraction of $c$.

For the same two ticks of a clock moving uniformly relative to a laboratory, the laboratory assigns an interval of 23.9 h while the moving clock records 24.0 h. Are these measurements consistent with special relativity? Explain without first calculating a speed.

Alice and Bob move past one another at constant relative velocity, each carrying a clock. After accounting for light-travel time, can each measure the other’s clock as running slow? Explain why this is not a contradiction, referring to how the two frames compare distant clocks at the same time.

How could a traveler return to Earth having aged much less than people who remained there? Would this also allow the traveler to return to Earth’s past? Explain.

(a) Which frame measures an object’s proper length? How does its length measured in a frame where it moves along its length compare with its proper length? In which frame is this length greatest? (b) A rod moves to the right past a row of laboratory rulers. One student records the position of its left end, and another records the position of its right end. They subtract the two positions and report the result as the rod’s length in the laboratory frame. What information about how the measurements were made is missing? Explain how their procedure must be arranged to measure the rod’s length, and how recording the right end later than the left end would affect their result.

A rectangular spacecraft moves horizontally relative to a laboratory. Its length is parallel to the motion and its height is perpendicular to it. Which dimension does the laboratory measure as contracted? Does an astronaut at rest inside measure either dimension as contracted? Explain why “all moving objects get smaller in every direction” is incorrect.

Alice and Bob each carry a meterstick parallel to their relative motion. Their speed relative to one another is $0.800c$. How long does Alice measure Bob’s stick to be? How long does Bob measure Alice’s? How can each observer measure the other’s stick as shorter than their own without a contradiction?

An astronaut measures the proper length of her spacecraft as 25.0 m. An observer in a frame where the spacecraft moves along its length reports 100 m. Can both be valid length measurements? Explain without solving for a speed.

Why are time dilation and length contraction hard to notice in everyday travel even though relativity applies to cars and airplanes? Explain in terms of $\gamma$ and the recovery of classical predictions at low speeds.

A spacecraft has a proper length of 200 m along its motion and passes Earth at $0.970c$. What length does an Earth-frame observer measure? Does someone on board measure that same contracted length?

A muon moves through the laboratory at $0.950c$ and has a proper mean lifetime of 2.20 μs. (a) Find its mean lifetime and mean travel distance in the laboratory. (b) In the muon’s frame it stays at rest; how far does the laboratory move during that proper lifetime? ($c$) Verify the length-contraction relationship between the two distances. Use $c=3.00\times {10}^8\text{m/s}$.

A truck moves at 30.0 m/s relative to the ground. A baseball pitcher standing in its bed throws a ball forward at 20.0 m/s relative to the truck. Use $u=\frac{v+u^{\prime }}{1+\frac{v{u}^{\prime }}{c^2}}$ with $c=3.00\times {10}^8\text{m/s}$. Why does your result agree with 50.0 m/s to ordinary measurement precision, while simply adding velocities is unreliable for a spacecraft moving at a substantial fraction of $c$?

Substitute $u^{\prime }=c$ into the relativistic velocity-addition formula and simplify. Repeat for $u^{\prime }=-c$. Explain how these results agree with Einstein’s second postulate for any frame speed $\left|v\right|<c$.

A spacecraft moves toward Earth at $0.750c$. It launches a canister at $0.500c$ relative to itself, first toward Earth and then, in a separate trial, away from Earth. Choose motion toward Earth as positive. Find the canister’s Earth-frame velocity in each trial. Does the backward launch necessarily make it move away from Earth?

For light emitted by a source moving directly away from an observer, is the observed wavelength longer or shorter than the emitted wavelength? What changes if the source approaches, or if their relative velocity is zero? Name the shifts.

Relativistic momentum is $p=\gamma mv$. (a) For an isolated system, does the law of conservation of momentum still apply? (b) What does this expression become when $v$ is much smaller than $c$? (c) For the same mass and velocity, what is the ratio of relativistic momentum to $mv$ when $\gamma =2$?

After a nuclear reaction, the separated products can have less combined rest mass than the initial particles. How can energy still be conserved? Does releasing energy in a chemical reaction also correspond to a mass change, even if that change is much harder to measure?

An electron has mass $9.11\times {10}^{\mathrm{-31}}\text{kg}$. (a) Calculate its rest energy in joules using $c=3.00\times {10}^8\text{m/s}$. (b) Express this rest energy in MeV, using $1\text{MeV}=1.602\times {10}^{\mathrm{-13}}\text{J}$. (c) If its Lorentz factor is 2.00, express its total energy and kinetic energy as multiples of that rest energy.

As the speed of an object with nonzero rest mass approaches $c$, what happens to $\gamma$, its momentum, and its total energy? Why does adding energy not allow it to reach or exceed $c$?

A photon has zero rest mass. Does that mean it must have zero energy and zero momentum? Use $E^2={(pc)}^2+{(m{c}^2)}^2$ to explain.