Special Relativity

Is Earth an inertial frame of reference? Is the Sun? Justify your response.

Does motion affect the rate of a clock as measured by an observer moving with it? Does motion affect how an observer moving relative to a clock measures its rate?

To whom does the elapsed time for a process seem to be longer, an observer moving relative to the process or an observer moving with the process? Which observer measures proper time?

How could you travel far into the future without aging significantly? Could this method also allow you to travel into the past?

To whom does an object seem greater in length, an observer moving with the object or an observer moving relative to the object? Which observer measures the object’s proper length?

Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?

Suppose an astronaut is moving relative to the Earth at a significant fraction of the speed of light. (a) Does he observe the rate of his clocks to have slowed? (b) What change in the rate of Earth-bound clocks does he see? (c) Does his ship seem to him to shorten? (d) What about the distance between stars that lie on lines parallel to his motion? (e) Do he and an Earth-bound observer agree on his velocity relative to the Earth?

Explain the meaning of the terms “red shift” and “blue shift” as they relate to the relativistic Doppler effect.

What happens to the relativistic Doppler effect when relative velocity is zero? Is this the expected result?

Is the relativistic Doppler effect consistent with the classical Doppler effect in the respect that \({{\rm{\lambda }}_{{\rm{obs}}}}\) is larger for motion away?

All galaxies farther away than about \({\rm{50 \times 1}}{{\rm{0}}^{\rm{6}}}{\rm{ ly}}\) exhibit a red shift in their emitted light that is proportional to distance, with those farther and farther away having progressively greater red shifts. What does this imply, assuming that the only source of red shift is relative motion? (Hint: At these large distances, it is space itself that is expanding, but the effect on light is the same.)

How does modern relativity modify the law of conservation of momentum?

What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.

Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.

The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.

We know that the velocity of an object with mass has an upper limit of c. Is there an upper limit on its momentum? Its energy? Explain.

Given the fact that light travels at c, can it have mass? Explain.

If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon's surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)

What is γ? (a) if v=\({\bf{0}}.{\bf{250c}}\)? (b) If v=\({\bf{0}}.{\bf{500c}}\)

What is γ? (a) if v=\({\bf{0}}.{\bf{100c}}\)? (b) If v=\({\bf{0}}.{\bf{900c}}\)

Particles called π-mesons are produced by accelerator beams. If these particles travel at \({\rm{2}}{\rm{.70 \times 1}}{{\rm{0}}^{\rm{8}}}{\rm{\;m/s }}\) and live \({\rm{2}}{\rm{.60 \times 1}}{{\rm{0}}^{{\rm{ - 8}}}}\) when at rest relative to an observer, how long do they live as viewed in the laboratory? 

Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at 0.980c0.980c, and it lives 1.24×10^-8when at rest relative to an observer. How long does it live as you observe it?

A neutral π-meson is a particle that can be created by accelerator beams. If one such particle lives 1.40×10⁴⁸s  as measured in the laboratory, and  0.840×10^-18 s when at rest relative to an observer, what is its velocity relative to the laboratory?

A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2065s?

If relativistic effects are to be less than 1%, then γ must be less than 1.01. At what relative velocity is γ = 1.01?

If relativistic effects are to be less than 3%, then γ must be less than 1.03. At what relative velocity is γ = 1.03?

(a) Show that \[{\left({pc}\right)^2}/{\left({m{c^2}{\rm{}}} \right)^2}{\rm{ }} = {\rm{ }}{\gamma ^2}{\rm{ }} - {\rm{ }}1\]. This means that at large velocities \[pc>>m{c^2}\]. (b) Is \[E{\rm{}}\approx{\rm{}}pc\] when \[\gamma{\rm{}}={\rm{}}30.0\], as for the astronaut discussed in the twin paradox?

What is ꝩ for a proton having a mass energy of 938.3MeV accelerated through an effective potential of 1.0TV (teravolt) at Fermilab outside Chicago?

(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if ꝩ= 1.00 ×10⁵  for them? (b) What is their total energy (nearly the same as kinetic in this case) in GeV?

(a) Using data from Table \(7.1\), find the mass destroyed when the energy in a barrel of crude oil is released. 

(b) Given these barrels contain \(200\) litres and assuming the density of crude oil is\(750{\rm{ }}kg/{m^3}\), what is the ratio of mass destroyed to original mass, \(\Delta m/m\)?

(a) Calculate the energy released by the destruction of 1.00kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?

A Van de Graaff accelerator utilizes a \(50.0{\rm{ }}MV\) potential difference to accelerate charged particles such as protons. 

(a) What is the velocity of a proton accelerated by such a potential? 

(b) An electron?

Suppose you use an average of \(500{\rm{ }}kW.h\)of electric energy per month in your home. 

(a) How long would \(1.00{\rm{ }}g\) of mass converted to electric energy with an efficiency of \(38\% \) last you?

 (b) How many homes could be supplied at the \(500{\rm{ }}kW.h\) 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 \(1000{\rm{ }}MW\) of electric power? 

(b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is \({10^4}{\rm{ }}kg\) ?

Nuclear-powered rockets were researched for some years before safety concerns became paramount.

 (a) What dfraction 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 \(250{\rm{ }}km\), and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) 

(b) If the ship has a mass of \(1.00{\rm{ }} \times {\rm{ }}{10^5}{\rm{ }}kg\)\((100{\rm{ }}tonns)\), what total yield nuclear explosion in tons of TNT is needed?

The Sun produces energy at a rate of \(4.00{\rm{ }} \times {\rm{ }}{10^{26}}\,W\) by the fusion of hydrogen. 

(a) How many kilograms of hydrogen undergo fusion each second? 

(b) If the Sun is \(90.0{\rm{ }}\% \)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 draction of its mass will it have lost in the time found in part (b)?