Chapter 28 : Special Relativity - all with Video Solutions
Problems & Exercises
Section 28.2: Simultaneity And Time Dilation
(a) Find the value of for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that 24.0 h have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
Section 28.3: Length Contraction
(a) How far does the muon in Example 28.1 travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction .
A spaceship is heading directly toward the Earth at a velocity of . The astronaut on board claims that he can send a canister toward the Earth at relative to the Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
Section 28.4: Relativistic Addition of Velocities
(a) Suppose the speed of light were only 3000 m/s . A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1000 m/s. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.
If a galaxy moving away from the Earth has a speed of 1000 km/s and emits 656 nm light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?
A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.
(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy away is receding from us at , at what velocity relative to us must we send an exploratory probe to approach the other galaxy at , as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)
Section 28.5: Relativistic Momentum
Section 28.6: Relativistic Energy
There is approximately of energy available from fusion of hydrogen in the world’s oceans. (a) If of this energy were utilized, what would be the decrease in mass of the oceans? Assume that 0.08% of the mass of a water molecule is converted to energy during the fusion of hydrogen. (b) How great a volume of water does this correspond to? (c) Comment on whether this is a significant fraction of the total mass of the oceans.
A -meson is a particle that decays into a muon and a massless particle. The -meson has a rest mass energy of 139.6 MeV, and the muon has a rest mass energy of 105.7 MeV. Suppose the -meson is at rest and all of the missing mass goes into the muon’s kinetic energy. How fast will the muon move?
A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?
Suppose you use an average of of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the per month rate for one year by the energy from the described mass conversion?
(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is ?
Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket’s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of (100 tons), what total yield nuclear explosion in tons of TNT is needed?
The Sun produces energy at a rate of by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?
Test Prep for AP® Courses
Section 28.1: Eintein's Postulates
Problem 1 (AP)
Which of the following statements describes the Michelson- Morley experiment? c.
- The speed of light is independent of the motion of the source relative to the observer. d.
- The speed of light is different in different frames of reference.
- The speed of light changes with changes in the observer.
- The speed of light is dependent on the motion of the source.
Section 28.4: Relativistic Addition of Velocities
Problem 2 (AP)
What happens when velocities comparable to the speed of light are involved in an observation?
- Newton’s second law of motion, F = ma , governs the motion of the object.
- Newton’s second law of motion, F = ma , no longer governs the dynamics of the object.
- Such velocities cannot be determined mathematically.
- None of the above