Solved: When an objects speed increases by 5%, its momentum increases by a factor of 5

Why was the MichelsonMorley experiment a more sensitive test of motion through the ether than independent measurements of the speed of light in two perpendicular directions?

Why was it necessary to repeat the MichelsonMorley experiment throughout the year?

Whats special about the special theory of relativity?

Does relativity require that the speed of sound be the same for all observers? Why or why not?

Time dilation is sometimes described by saying that moving clocks run slow. In what sense is this true? In what sense does the statement violate the spirit of relativity?

If youre in a spaceship moving at 0.95c relative to Earth, do you perceive time to be passing more slowly than it would on Earth? Think! Is your answer consistent with the relativity principle?

The Andromeda Galaxy is 2 million light years from the Milky Way. Although nothing can go faster than light, it would still be possible to travel to Andromeda in much less than 2 million years. How?

Is matter converted to energy in a nuclear reactor? In a burning candle? In your body?

If you took your pulse while traveling in a high-speed spacecraft, would it be faster than, slower than, or the same as on Earth?

The rest energy of an electron is 511 keV. Whats the approximate speed of an electron whose total energy is 1 GeV? (Note: No calculations needed!)

An atom in an excited state emits a burst of light. What happens to the atoms mass?

The quantity E S # B S is invariant. What does this say about how different observers will measure the angle between E S and B S in a light wave?

An airplane makes a round trip between two points 1800 km apart, flying with airspeed 800 km/h. Whats the round trip flying time (a) if theres no wind, (b) with wind at 130 km/h perpendicular to a line joining the two points, and (c) with wind at 130 km/h along a line joining the two points?

Consider a MichelsonMorley experiment with 11-m light paths perpendicular and parallel to the ether wind. What would be the difference in light travel times on the two paths if Earth moved relative to the ether at (a) its orbital speed (Appendix E); (b) 0.01c; (c) 0.5c; and (d) 0.99c?

Two stars are 50 ly apart, measured in their common rest frame. How far apart are they to a spaceship moving between them at 0.75c?

How long would it take a spacecraft traveling at 0.65c to get from Earth to Pluto according to clocks (a) on Earth and (b) on the spacecraft? Assume Earth and Pluto are on the same side of the Sun.

A spaceship passes by you at half the speed of light, and you determine that its 35 m long. Find its length as measured in its rest frame.

An extraterrestrial spacecraft whizzes through the solar system at 0.80c. How long does it take to go the 8.3 light-minute-distance from Earth to Sun (a) according to an observer on Earth and (b) according to an alien aboard the ship?

How fast would you have to move relative to a meter stick for it to be 99 cm long in your reference frame?

A hospitals linear accelerator produces electron beams for cancer treatment. The accelerator is 1.6 m long and the electrons reach a speed of 0.98c. How long is the accelerator in the electrons reference frame?

By what factor does an objects momentum change if you double its speed when its original speed is (a) 25 m/s and (b) 100 Mm/s?

At what speed will the momentum of a proton (mass 1 u) equal that of an alpha particle (mass 4 u) moving at 0.5c?

At what speed will the Newtonian expression for momentum be in error by 1%?

A particle is moving at 0.90c. If its speed increases by 10%, by what factor does its momentum increase?

Find (a) the total energy and (b) the kinetic energy of an electron moving at 0.97c

At what speed will the relativistic and Newtonian expressions for kinetic energy differ by 10%?

Show that the time of Equation 33.2 is longer than that of Equation 33.1 when 0 6 v 6 c.

Youre designing a Michelson interferometer in which a speedof-light difference of 100 m/s in two perpendicular directions is supposed to shift the interference pattern so a bright fringe of 550-nm light ends up where the adjacent dark fringe would be in the absence of a speed difference. How long should you make the interferometers arms?

Earth and Sun are 8.3 light minutes apart, as measured in their rest frame. (a) Whats the speed of a spacecraft that makes the trip in 5.0 min according to its on-board clocks? (b) Whats the trip time as measured by clocks in the EarthSun reference frame?

Youre the communications officer on a fast spaceship that takes 50 years in ship time to reach the Andromeda Galaxy, 2 million light years from Earth in the common rest frame of Earth and Andromeda. As soon as you reach Andromeda, your captain orders you to send a radio message to Earth announcing your arrival; he claims the message will reach Earth about a century after you left. You claim it will be much later when the message arrives. Whos right?

You wish to travel to a star N light years from Earth. How fast must you go if the one-way journey is to occupy N years of your life?

The nearest star beyond our solar system is about 4 light years away. If a spaceship can get to the star in 5 years, as measured on Earth, (a) how long would the ships pilot judge the journey to take? (b) How far from Earth would the pilot find the star to be?

Twins A and B live on Earth. On their 20th birthday, twin B climbs into a spaceship and makes a round-trip journey at 0.95c to a star 30 light years distant, as measured in the Earthstar reference frame. What are their ages when twin B returns to Earth?

Radioactive oxygen-15 decays at such a rate that half the atoms in a given sample decay every 2 min. If a tube containing 1000 O-15 atoms is moved at 0.80c relative to Earth for 6.67 min according to clocks on Earth, how many atoms will be left at the end of that time?

Two distant galaxies are receding from Earth at 0.75c in opposite directions. How fast does an observer in one galaxy measure the other to be moving?

Two spaceships are racing. The slower one passes Earth at 0.70c, and the faster one moves at 0.40c relative to the slower one. Whats the faster ships speed relative to Earth?

Use relativistic velocity addition to show that if an object moves at speed v 6 c relative to some inertial reference frame, then its speed relative to any other inertial frame must also be less than c.

Earth and Sun are 8.33 light minutes apart. Event A occurs on Earth at time t = 0 and event B on the Sun at t = 2.45 min, as measured in the EarthSun frame. Find the time order and time difference between A and B for observers (a) moving on a line from Earth to Sun at 0.750c, (b) moving on a line from Sun to Earth at 0.750c, and (c) moving on a line from Earth to Sun at 0.294c.

Youre writing a galactic history involving two civilizations that evolve on opposite sides of a 1.0 *105 -ly-diameter galaxy. In the galaxys reference frame, civilization B launched its first spacecraft 45,000 years after civilization A. You and your readers, from a more advanced civilization, are traveling through the galaxy at 0.99c on a line from A to B. Which civilization do you record as having first achieved interstellar travel, and how much in advance of the other?

Repeat Problem 39, now assuming that civilization B lags A by 1.2 million years in the galaxys reference frame.

Could there be observers who would judge the two events in Problem 39 to be simultaneous? If so, how fast and in what direction must these observers be moving?

Could there be observers who would judge the two events in Problem 40 to be simultaneous? If so, how fast and in what direction must these observers be moving?

The Curiosity rover touched down on Mars when Earth and Mars were 14 light-minutes apart. At the instant of touchdown, clocks at Mission Control in Pasadena, California, read 10:31 pm. As judged by observers on a spacecraft heading along the Earth Mars line at 0.35c, did touchdown occur before or after the time the clocks in Pasadena read 10:31 pm, and by how much?

Derive the Lorentz transformations for time from the transformations for space.

In the light box of Fig. 33.6, let event A be the emission of the light flash and event B its return to the source. Assign suitable space and time coordinates to these events in the frame in which the box moves with speed v. Apply the Lorentz transformations to show that the time between the two events in the box frame is given by Equation 33.3.

Youre a consultant for the director of a sci-fi movie. The film starts with two spaceships, each measuring 25 m long in its rest frame, approaching Earth in opposite directions with speeds shown in Fig. 33.24. The director wants to know how long to make ship B for scenes shot (a) in Earths reference frame and (b) in ship As frame. Your answers? A B v = 0.65c v = 0.50c

How fast would you have to go to reach a star 240 light years away in an 85-year human lifetime?

An advanced civilization has developed a spaceship that goes, with respect to the galaxy, only 50 km/s slower than light. (a) According to the ships crew, how long does it take to cross the galaxys 100,000-ly diameter? (b) Whats the galactic diameter measured in the ships reference frame?

distant, as measured in the Earthstar reference frame. Let event A be the ships departure from Earth and event B its arrival at the star. (a) Find the distance and time between the two events in the Earthstar frame. (b) Repeat for the ships frame. (Hint: The distance in the ship frame is the distance an observer has to move with respect to that frame to be at both eventsnot the same as the Lorentz-contracted distance between Earth and star.) (c) Compute the square of the spacetime interval in both frames to show explicitly that its invariant

Use Equation 33.6 to calculate the square of the spacetime interval between the events (a) of Problem 39 and (b) of Problem 40. Comment on the signs of your answers in relation to the possibility of a causal relationship between the events

A light beam is emitted at event A and arrives at event B. Show that the spacetime interval between the two events is zero.

Compare the momentum changes needed to boost a spacecraft (a) from 0.10c to 0.20c and (b) from 0.80c to 0.90c.

Event A occurs at x = 0 and t = 0 in reference frame S. Event B occurs at x = 3.8 light years and t = 1.6 years in S. Find (a) the distance and (b) the time between A and B in a frame moving at 0.80c along the x-axis of S.

When a particles speed doubles, its momentum increases by a factor of 3. What was its original speed?

Find (a) the speed and (b) the momentum of a proton with kinetic energy 500 MeV

The Large Hadron Collider accelerates protons to energies of 14 TeV. (a) Compare that proton energy with the kinetic energy of a 25-mg bug crawling at 2.0 mm/s. (b) Compare the proton momentum with that of the same bug.

A large city consumes electrical energy at the rate of 1 GW. If you converted all the rest mass in a 1-g raisin to electrical energy, for how long could it power the city?

In a nuclear-fusion reaction, two deuterium nuclei combine to make a helium nucleus plus a neutron, releasing 3.3 MeV of energy in the process. By how much do the combined masses of the helium nucleus and the neutron differ from the combined masses of the original deuterium nuclei?

Find the kinetic energy of an electron moving at (a) 0.0010c, (b) 0.60c, and (c) 0.99c. Use suitable approximations where possible.

Find the speed of an electron with kinetic energy (a) 100 eV, (b) 100 keV, (c) 1 MeV, and (d) 1 GeV. Use suitable approximations where possible.

Use the binomial approximation (Appendix A) to show that Equation 33.8 reduces to the Newtonian expression for kinetic energy in the limit u V c.

Show that Equation 33.10 follows from the expressions for relativistic momentum and total energy

Show from the Lorentz transformations that the spacetime interval of Equation 33.6 has the same value in all reference frames

How fast would you have to travel to reach the Crab Nebula, 6500 light years from Earth, in 15 years? Give your answer to seven significant figures

At what speed are a particles kinetic and rest energies equal?

The highest-energy cosmic rays ever detected are protons with energies on the order of 300 EeV. (a) Whats Earths radius as measured in the reference frame of such a proton as it approaches Earth? (b) Compare the high-energy protons total energy with that of a 143-g baseball moving at 100 km/h. (c) Compare the protons momentum with that of the same baseball.

When an objects speed increases by 5%, its momentum increases by a factor of 5. What was its original speed?

Use the Lorentz transformations to show that if two events are separated in space and time so that a light signal leaving one event cannot reach the other, then there is an observer for whom the two events are simultaneous. Show that the converse is also true: If a light signal can get from one event to the other, then no observer will find them simultaneous

A source emitting light with frequency f moves toward you at speed u. By considering both time dilation and the effect of wavefronts piling up as shown in Fig. 14.33, show that you measure a Doppler-shifted frequency given by f = f A c + u c - u Use the binomial approximation (Appendix A) to show that this result can be written in the form of Equation 14.15 when u V c.

Equation 33.5a transforms the velocity u ! of an object moving in the x-directionthe same direction as the relative velocity v ! of the two reference frames. Now suppose the objects velocity also has a component uy perpendicular to the two frames relative velocity v ! . Find the transformation from u y to uy.

Consider a relativistic particle of mass m moving along a straight line. Use Equation 33.7 to find an expression for the force on the particle, defined as F = dp/dt, in terms of its acceleration a = du/dt

Find the speed of a particle whose relativistic kinetic energy is 50% greater than the Newtonian value calculated for the same speed

Its the 24th century, and youre a curator at the Starfleet Museum of Ancient Technology. Archaeologists have unearthed a TV tube, an ancient device for displaying moving images. Your job is to get it working. One reference says the device accelerated electrons, which then bombarded a screen to produce images; to the electrons, the tube was 57 cm long. You measure the tube and find its 60 cm long. To get it working, you need to know the electrons speed and the potential difference needed to accelerate them. The electrons rest energy is 511 keV. Your answers?

Consider a line of positive charge with line charge density l as measured in a frame S at rest with respect to the charges. (a) Show that the electric field a distance r from this charged line has magnitude E = l/2pP0r, and that theres no magnetic field (no relativity needed here). Now consider the situation in a frame S moving at speed v parallel to the line of charge. (b) Show that the charge density measured in S is given by l = gl, where g = 1/21 - v2 /c2 . (c) Use the result of (b) to find the electric field in S. Since the charge is moving with respect to S, theres a current in S. (d) Find an expression for this current and (e) for the magnetic field it produces. Determine the values of the quantities (f) E S # B S and (g) E2 - c2 B2 in both reference frames, and show that these quantities are invariant. Your result gives a hint

The table below lists the total energy and corresponding momentum for a particle. Theyre measured in MeV and MeV/c, respectivelycommonly used units in particle physics. Determine suitable functions of these quantities to plot such that the resulting plot should be a straight line. Plot your data, determine a best-fit line, and use it to find (a) the value of c and (b) the particles mass. You may need to convert to SI before plotting. Can you identify the particle? Total energy, E (MeV) 0.511 1.01 1.51 2.51 3.51 4.51 5.51 Momentum, p (MeV/c) 0 0.872 1.41 2.46 3.45 4.61 5.49 Passage Problems Youve been named captain of NASAs first interstellar mission since the Voyager robotic spacecraft. You board your spaceship, accelerate quickly to 0.8c, and cruise at constant speed toward Proxima Centauri, the closest star to our Sun. Proxima Centauri is 4 light-years distant as measured in the two stars common rest frame. On the way, you conduct various medical experiments to determine the effects of a long space voyage on the human body.

Taking your pulse, you find a. its significantly slower than when youre on Earth. b. its the same as when youre on Earth. c. its significantly faster than when youre on Earth

How much do you age during your interstellar journey? a. 3 years b. just under 4 years c. just over 4 years d. 5 years

Back on Earth, Mission Control judges that your shipboard clocks run slow. What do you judge about clocks at Mission Control? a. They run fast. b. They keep time at the same rate as your clocks. c. They run slow. d. You cant tell anything about their clocks.

In your spaceships reference frame, the distance from the Sun to Proxima Centauri is a. 2.4 light years. b. just under 4 light years. c. 4 light years. d. 5 light years.