Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
Before doing any calculations involving time dilation or length contraction, it is useful to identify which observer measures the proper time interval or the proper length L0.
ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.

Section 28.3 The Relativity of Time: Time Dilation

| 1 | ssm A Klingon spacecraft has a speed of 0.75c with respect to the earth. The Klingons measure 37.0 h for the time interval be tween two events on the earth. What value for the time interval would they measure if their ship had a speed of 0.94c with respect to the earth?

| 2 | A particle known as a pion lives for a short time before breaking apart into other particles. Suppose a pion is moving at a speed of 0.990c, and an observer who is stationary in a laboratory measures the pion’s lifetime to be . (a) What is the lifetime according to a hypothetical person who is riding along with the pion? (b) According to this hypothetical person, how far does the laboratory move before the pion breaks apart?

| 3 | ssm A law enforcement officer in an intergalactic “police car” turns on a red flashing light and sees it generate a flash every 1.5 s. A person on earth measures that the time between flashes is 2.5 s. How fast is the “police car” moving relative to the earth?

| 4 | Suppose you are planning a trip in which a spacecraft is to travel at a constant velocity for exactly six months, as measured by a clock on board the spacecraft, and then return home at the same speed. Upon return, the people on earth will have advanced exactly one hundred years into the future. According to special relativity, how fast must you travel? Express your answer to 5 significant figures as a multiple of c—for example, 0.955 85c.

* | 5 | Interactive Solution 28.5 at illustrates one way to model this problem. A 6.00-kg object oscillates back and forth at the end of a spring whose spring constant is 76.0 N/m. An observer is traveling at a speed of relative to the fixed end of the spring. What does this observer measure for the period of oscillation?

* | 6 | An astronaut travels at a speed of 7800 m/s relative to the earth, a speed that is very small compared to c. According to a clock on the earth, the trip lasts 15 days. Determine the difference (in seconds) between the time recorded by the earth clock and the astronaut’s clock. [Hint: When , the following approximation is valid: .]

** | 7 | As observed on earth, a certain type of bacterium is known to double in number every 24.0 hours. Two cultures of these bacteria are prepared, each consisting initially of one bacterium. One culture is left on earth and the other placed on a rocket that travels at a speed of 0.866c relative to the earth. At a time when the earthbound culture has grown to 256 bacteria, how many bacteria are in the culture on the rocket, according to an earth-based observer?

Section 28.4 The Relativity of Length: Length Contraction

| 8 | A tourist is walking at a speed of 1.3 m/s along a 9.0-km path that follows an old canal. If the speed of light in a vacuum were 3.0 m/s, how long would the path be, according to the tourist?

| 9 | ssm www Two spaceships A and B are exploring a new planet. Relative to this planet, spaceship A has a speed of 0.60c, and space ship B has a speed of 0.80c. What is the ratio DA/DB of the values for the planet’s diameter that each spaceship measures in a direction that is parallel to its motion?

| 10 | Interactive LearningWare 28.1 at reviews the concepts that play roles in this problem. The distance from earth to the center of our galaxy is about 23 000 ly , as measured by an earth-based observer. A spaceship is to make this journey at a speed of 0.9990c. According to a clock on board the spaceship, how long will it take to make the trip? Express your answer in years .

| 11 | ssm A UFO streaks across the sky at a speed of 0.90c relative to the earth. A person on earth determines the length of the UFO to be 230 m along the direction of its motion. What length does the person measure for the UFO when it lands?

| 12 | An unstable high-energy particle is created in the laboratory, and it moves at a speed of 0.990c. Relative to a stationary reference frame fixed to the laboratory, the particle travels a distance of 1.05 × 10−3 m before disintegrating. What is (a) the proper distance and (b) the distance measured by a hypothetical person traveling with the particle? Determine the particle’s (c) proper lifetime and (d) its dilated lifetime.

| 13 | Interactive Solution 28.13 at illustrates one approach to solving this problem. A space traveler moving at a speed of 0.70c with respect to the earth makes a trip to a distant star that is stationary relative to the earth. He measures the length of this trip to be 6.5 light-years. What would be the length of this same trip (in light-years) as measured by a traveler moving at a speed of 0.90c with respect to the earth?

* | 14 | As the drawing shows, a carpenter on a space station has constructed a 30.0° ramp. A rocket moves past the space station with a relative speed of 0.730c in a direction parallel to side x. What does a person aboard the rocket measure for the angle of the ramp?

** | 15 | Twins who are 19.0 years of age leave the earth and travel to a distant planet 12.0 light-years away. Assume that the planet and earth are at rest with respect to each other. The twins depart at the same time on different spaceships. One twin travels at a speed of 0.900c, and the other twin travels at 0.500c. (a) According to the theory of special relativity, what is the difference between their ages when they meet again at the earliest possible time? (b) Which twin is older?

** | 16 | A woman and a man are on separate rockets, which are flying parallel to each other and have a relative speed of 0.940c. The woman measures the same value for the length of her own rocket and for the length of the man’s rocket. What is the ratio of the value that the man measures for the length of his own rocket to the value he measures for the length of the woman’s rocket?
Section 28.5 Relativistic Momentum

| 17 | A spaceship is approaching the earth at a relative speed of 0.85c. The mass of the ship is 2.0 × 107 kg. Find the magnitude of (a) the classical momentum and (b) the relativistic momentum of the ship.

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