The rest energy E0 and the total energy E of three

m A particle known as a pion lives for a short time before breaking apart into other particles. Suppose that a pion is moving at a speed of 0.990c, and an observer who is stationary in a laboratory measures the pions lifetime to be 3.5 3 1028 s. (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?

A radar antenna is rotating and makes one revolution every 25 s, as measured on earth. However, instruments on a spaceship moving with respect to the earth at a speed v measure that the antenna makes one revolution every 42 s. What is the ratio v/c of the speed v to the speed c of light in a vacuum?

Suppose that 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 your 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 fi ve signifi cant fi gures as a multiple of cfor example, 0.955 85c

Suppose that you are traveling on board a spacecraft that is moving with respect to the earth at a speed of 0.975c. You are breathing at a rate of 8.0 breaths per minute. As monitored on earth, what is your breathing rate?

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 1.90 3 108 m/s relative to the fi xed end of the spring. What does this observer measure for the period of oscillation?

A spaceship travels at a constant speed from earth to a planet orbiting another star. When the spacecraft arrives, 12 years have elapsed on earth, and 9.2 years have elapsed on board the ship. How far away (in meters) is the planet, according to observers on earth?

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 earthbased observer?

Suppose the straight-line distance between New York and San Francisco is 4.1 3 106 m (neglecting the curvature of the earth). A UFO is fl ying between these two cities at a speed of 0.70c relative to the earth. What do the voyagers aboard the UFO measure for this distance?

How fast must a meter stick be moving if its length is observed to shrink to one-half of a meter?

The distance from earth to the center of our galaxy is about 23 000 ly (1 ly 5 1 light-year 5 9.47 3 1015 m), 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 (1 yr 5 3.16 3 107 s).

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?

A Martian leaves Mars in a spaceship that is heading to Venus. On the way, the spaceship passes earth with a speed v 5 0.80c relative to it. Assume that the three planets do not move relative to each other during the trip. The distance between Mars and Venus is 1.20 3 1011 m, as measured by a person on earth. (a) What does the Martian measure for the distance between Mars and Venus? (b) What is the time of the trip (in seconds) as measured by the Martian?

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

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 fi xed to the laboratory, the particle travels a distance of 1.05 3 1023 m before disintegrating. What are (a) the proper distance and (b) the distance measured by a hypothetical person traveling with the particle? Determine the particles (c) proper lifetime and (d) its dilated lifetime

As the drawing shows, a carpenter on a space station has constructed a 30.08 ramp. A rocket moves past the space station with a relative speed of 0.730c in a direction parallel to side x0. What does a person aboard the rocket measure for the angle of the ramp?

An object is made of glass and has the shape of a cube 0.11 m on a side, according to an observer at rest relative to it. However, an observer moving at high speed parallel to one of the objects edges and knowing that the objects mass is 3.2 kg determines its density to be 7800 kg/m3 , which is much greater than the density of glass. What is the moving observers speed (in units of c) relative to the cube?

A rectangle has the dimensions of 3.0 m 3 2.0 m when viewed by someone at rest with respect to it. When you move past the rectangle along one of its sides, the rectangle looks like a square. What dimensions do you observe when you move at the same speed along the adjacent side of the rectangle?

At what speed is the magnitude of the relativistic momentum of a particle three times the magnitude of the nonrelativistic momentum?

What is the magnitude of the relativistic momentum of a proton with a relativistic total energy of 2.7 3 10210 J?

A spacecraft has a nonrelativistic (or classical) momentum whose magnitude is 1.3 3 1013 kg ?m/s. The spacecraft moves at such a speed that the pilot measures the proper time interval between two events to be one-half the dilated time interval. Find the relativistic momentum of the spacecraft.

A woman is 1.6 m tall and has a mass of 55 kg. She moves past an observer with the direction of the motion parallel to her height. The observer measures her relativistic momentum to have a magnitude of 2.0 3 1010 kg ?m/s. What does the observer measure for her height?

Three particles are listed in the table. The mass and speed of each particle are given as multiples of the variables m and v, which have the values m 5 1.20 3 1028 kg and v 5 0.200c. The speed of light in a vacuum is c 5 3.00 3 108 m/s. Determine the momentum for each particle according to special relativity

Starting from rest, two skaters push off against each other on smooth level ice, where friction is negligible. One is a woman and one is a man. The woman moves away with a velocity of 12.5 m/s relative to the ice. The mass of the woman is 54 kg, and the mass of the man is 88 kg. Assuming that the speed of light is 3.0 m/s, so that the relativistic momentum must be used, fi nd the recoil velocity of the man relative to the ice. (Hint: This problem is similar to Example 6 in Chapter 7.)

Radium is a radioactive element whose nucleus emits an particle (a helium nucleus) with a kinetic energy of about 7.8 3 10213 J (4.9 MeV). To what amount of mass is this energy equivalent?

How much work must be done on an electron to accelerate it from rest to a speed of 0.990c?

Review Conceptual Example 9 for background pertinent to this problem. Suppose that the speed of light in a vacuum were one million times smaller than its actual value, so that c 5 3.00 3 102 m/s. The spring constant of a spring is 850 N/m. Determine how far you would have to compress the spring from its equilibrium length in order to increase its mass by 0.010 g

Suppose that one gallon of gasoline produces 1.1 3 108 J of energy, and this energy is suffi cient to operate a car for twenty miles. An aspirin tablet has a mass of 325 mg. If the aspirin could be converted completely into thermal energy, how many miles could the car go on a single tablet?

Two kilograms of water are changed (a) from ice at 0 8C into liquid water at 0 8C and (b) from liquid water at 100 8C into steam at 100 8C. For each situation, determine the change in mass of the water.

Determine the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy (1 2 mv2 ) when a particle has a speed of (a) 1.00 3 1023 c and (b) 0.970c

Multiple-Concept Example 6 reviews the principles that play a role in this problem. A nuclear power reactor generates 3.0 3 109 W of power. In one year, what is the change in the mass of the nuclear fuel due to the energy being taken from the reactor?

Multiple-Concept Example 6 explores the approach taken in problems such as this one. Quasars are believed to be the nuclei of galaxies in the early stages of their formation. Suppose that a quasar radiates electromagnetic energy at the rate of 1.0 3 1041 W. At what rate (in kg/s) is the quasar losing mass as a result of this radiation?

An electron is accelerated from rest through a potential diff erence that has a magnitude of 2.40 3 107 V. The mass of the electron is 9.11 3 10231 kg, and the negative charge of the electron has a magnitude of 1.60 3 10219 C. (a) What is the relativistic kinetic energy (in joules) of the electron? (b) What is the speed of the electron? Express your answer as a multiple of c, the speed of light in a vacuum

An object has a total energy of 5.0 3 1015 J and a kinetic energy of 2.0 3 1015 J. What is the magnitude of the objects relativistic momentum?

You are driving down a two-lane country road, and a truck in the opposite lane is traveling toward you. Suppose that the speed of light in a vacuum is c 5 65 m/s. Determine the speed of the truck relative to you when (a) your speed is 25 m/s and the trucks speed is 35 m/s and (b) your speed is 5.0 m/s and the trucks speed is 55 m/s. The speeds given in parts (a) and (b) are relative to the ground.

A spacecraft approaching the earth launches an exploration vehicle. After the launch, an observer on earth sees the spacecraft approaching at a speed of 0.50c and the exploration vehicle approaching at a speed of 0.70c. What is the speed of the exploration vehicle relative to the spaceship?

Spaceships of the future may be powered by ion-propulsion engines in which ions are ejected from the back of the ship to drive it forward. In one such engine the ions are to be ejected with a speed of 0.80c relative to the spaceship. The spaceship is traveling away from the earth at a speed of 0.70c relative to the earth. What is the velocity of the ions relative to the earth? Assume that the direction in which the spaceship is traveling is the positive direction, and be sure to assign the correct plus or minus signs to the velocities.

The spaceship Enterprise 1 is moving directly away from earth at a velocity that an earth-based observer measures to be 10.65c. A sister ship, Enterprise 2, is ahead of Enterprise 1 and is also moving directly away from earth along the same line. The velocity of Enterprise 2 relative to Enterprise 1 is 10.31c. What is the velocity of Enterprise 2, as measured by the earth-based observer?

A person on earth notices a rocket approaching from the right at a speed of 0.75c and another rocket approaching from the left at 0.65c. What is the relative speed between the two rockets, as measured by a passenger on one of them?

Refer to Conceptual Example 11 as an aid in solving this problem. An intergalactic cruiser has two types of guns: a photon cannon that fi res a beam of laser light and an ion gun that shoots ions at a velocity of 0.950c relative to the cruiser. The cruiser closes in on an alien spacecraft at a velocity of 0.800c relative to this spacecraft. The captain fi res both types of guns. At what velocity do the aliens see (a) the laser light and (b) the ions approach them? At what velocity do the aliens see (c) the laser light and (d) the ions move away from the cruiser?

Two identical spaceships are under construction. The constructed length of each spaceship is 1.50 km. After being launched, spaceship A moves away from earth at a constant velocity (speed is 0.850c) with respect to the earth. Spaceship B follows in the same direction at a diff erent constant velocity (speed is 0.500c) with respect to the earth. Determine the length that a passenger on one spaceship measures for the other spaceship.

Two atomic particles approach each other in a head-on collision. Each particle has a mass of 2.16 3 10225 kg. The speed of each particle is 2.10 3 108 m/s when measured by an observer standing in the laboratory. (a) What is the speed of one particle as seen by the other particle? (b) Determine the magnitude of the relativistic momentum of one particle, as it would be observed by the other.

An electron and a positron have masses of 9.11 3 10231 kg. They collide and both vanish, with only electromagnetic radiation appearing after the collision. If each particle is moving at a speed of 0.20c relative to the laboratory before the collision, determine the energy of the electromagnetic radiation.

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?

The speed of an ion in a particle accelerator is doubled from 0.460c to 0.920c. The initial relativistic momentum of the ion is 5.08 3 10217 kg ?m/s. Determine (a) the mass and (b) the magnitude of the fi nal relativistic momentum of the ion.

The speed of an ion in a particle accelerator is doubled from 0.460c to 0.920c. The initial relativistic momentum of the ion is 5.08 3 10217 kg ?m/s. Determine (a) the mass and (b) the magnitude of the fi nal relativistic momentum of the ion.

An unstable particle is at rest and suddenly breaks up into two fragments. No external forces act on the particle or its fragments. One of the fragments has a velocity of 10.800c and a mass of 1.67 3 10227 kg, and the other has a mass of 5.01 3 10227 kg. What is the velocity of the more massive fragment? (Hint: This problem is similar to Example 6 in Chapter 7.)

The crew of a rocket that is moving away from the earth launches an escape pod, which they measure to be 45 m long. The pod is launched toward the earth with a speed of 0.55c relative to the rocket. After the launch, the rockets speed relative to the earth is 0.75c. What is the length of the escape pod as determined by an observer on earth?

The table gives the total energy and the rest energy for three objects in terms of an energy increment . For each object, determine the speed as a multiple of the speed c of light in a vacuum.

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 diff erent 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 diff erence between their ages when they meet again at the earliest possible time? (b) Which twin is older?

Imagine playing golf in a world where the speed of light is only c 5 3.40 m/s. Golfer A drives a ball down a fl at horizontal fairway for a distance that he measures as 75.0 m. Golfer B, riding in a cart, happens to pass by just as the ball is hit (see the fi gure). Golfer A stands at the tee and watches while golfer B moves down the fairway toward * the ball at a constant speed of 2.80 m/s. Concepts: (i) Who measures the proper length of the drive, and who measures the contracted length? (ii) Who measures the proper time interval, and who measures the dilated time interval? Calculations: (a) How far is the ball hit according to golfer B? (b) According to each golfer, how much time does it take golfer B to reach the ball?

The rest energy E0 and the total energy E of three particles, expressed in terms of a basic amount of energy E 5 5.98 3 10210 J, are listed in the table. The speeds of these particles are large, in some cases approaching the speed of light. Concepts: (i) Given the rest energies specifi ed in the table, what is the ranking (largest fi rst) of the masses of the particles? (ii) Is the kinetic energy KE given by the expression KE 5 1 /2mv2 , and what is the ranking (largest fi rst) of the kinetic energies of the particles? Calculations: For each particle, determine its (a) mass and (b) kinetic energy.