## Section 52.15 Special Relativity Bootcamp

### Subsection 52.15.1 Galilean Relativity

###### Problem 52.15.1. Velocity of a Ball Observed from Station and from Inside a Moving Train.

Follow the link: Checkpoint 52.1.4.

###### Problem 52.15.2. Speed of Mass Realtive to Earth Given Their Velocities with Respect to the Sun.

Follow the link: Checkpoint 52.1.5.

###### Problem 52.15.3. Speed of a Runner with Respect to a Train.

Follow the link: Checkpoint 52.1.6.

###### Problem 52.15.4. Speed of a Runner with Respect to a Train Example 2.

Follow the link: Checkpoint 52.1.8.

###### Problem 52.15.5. Momentum Conservation from the Perspective of Two Frames.

Follow the link: Checkpoint 52.1.10.

###### Problem 52.15.6. Momentum Conservation from the Perspective of Two Frames Example 2.

Follow the link: Checkpoint 52.1.11.

### Subsection 52.15.2 Time Dilation

###### Problem 52.15.7. Neutron lifetime.

Follow the link: Checkpoint 52.6.3.

###### Problem 52.15.8. Travel time.

Follow the link: Checkpoint 52.6.4.

###### Problem 52.15.9. Time Dilation of Clock in a Spaceship.

Follow the link: Checkpoint 52.6.6.

###### Problem 52.15.10. Mystery of muons.

Follow the link: Checkpoint 52.6.5.

###### Problem 52.15.11. Heartbeat Rate of an Astronaut in Two Different Frames.

Follow the link: Checkpoint 52.6.7.

###### Problem 52.15.12. Relation of Duration in Two Spaceships in Relative Motion.

Follow the link: Checkpoint 52.6.8.

### Subsection 52.15.3 Lorentz transformations

###### Problem 52.15.13. Describing Events in a Frame.

Follow the link: Checkpoint 52.7.4.

###### Problem 52.15.14. Coordinates and Times Assigned to an Event Described in another Frame.

Follow the link: Checkpoint 52.7.5.

###### Problem 52.15.15. Same Events Described by Observers in Two Frames.

Follow the link: Checkpoint 52.7.6.

### Subsection 52.15.4 Length Contraction

###### Problem 52.15.16. Relation of Distance in Two Spaceships in Relative Motion.

Follow the link: Checkpoint 52.8.2.

###### Problem 52.15.17. Size of a spaceship.

Follow the link: Checkpoint 52.8.3.

###### Problem 52.15.18. Relation of Duration and Distance in Two Spaceships in Relative Motion.

Follow the link: Checkpoint 52.8.4.

### Subsection 52.15.5 Transformations of velocities

###### Problem 52.15.19. Speed of Light in Two Frames.

Follow the link: Checkpoint 52.10.1.

###### Problem 52.15.20. Speed of a Spaceship from Two Frames.

Follow the link: Checkpoint 52.10.2.

###### Problem 52.15.21. Speed of a Proton Shot from a Spaceship.

Follow the link: Checkpoint 52.10.3.

###### Problem 52.15.22. Speed of an Electron Shot from a Spaceship.

Follow the link: Checkpoint 52.10.4.

###### Problem 52.15.23. Speeds of Photons Moving in Opposite Directions.

Follow the link: Checkpoint 52.10.5.

###### Problem 52.15.24. Colliding Spaceships Moving Perpendicularly to Each Other.

Follow the link: Checkpoint 52.10.6.

### Subsection 52.15.6 The Relativisitc Doppler Effect

###### Problem 52.15.25. Doppler Shift in Astronomy.

Follow the link: Checkpoint 52.11.2.

###### Problem 52.15.26. Recession Speed of a Star from Wavelength of its Lyman Line.

Follow the link: Checkpoint 52.11.6.

###### Problem 52.15.27. Approximate Doppler Effect from a Speeding Car.

Follow the link: Checkpoint 52.11.7.

###### Problem 52.15.28. Shift in Frequency of Light Due to Motion of the Detector.

Follow the link: Checkpoint 52.11.8.

###### Problem 52.15.29. Redshift of a Galaxy from Doppler Effect.

Follow the link: Checkpoint 52.11.9.

###### Problem 52.15.30. Speed of a Quasar from its Redshift.

Follow the link: Checkpoint 52.11.10.

### Subsection 52.15.7 Relativistic Momentum

###### Problem 52.15.31. Relativistic Equation of Motion of an Electron in a Constant Electric Field.

Follow the link: Checkpoint 52.12.2.

###### Problem 52.15.32. Equation of Motion of an Electron in a Cyclotron.

Follow the link: Checkpoint 52.12.3.

###### Problem 52.15.33. Relativisitc Momentum Versus Ordinary Momentum of an Electron.

Follow the link: Checkpoint 52.12.4.

###### Problem 52.15.34. Relativisitic Momentum of a Proton in Lab Frame.

Follow the link: Checkpoint 52.12.5.

###### Problem 52.15.35. Error When Using Non-Relativisitc Formula for Momentum.

Follow the link: Checkpoint 52.12.6.

###### Problem 52.15.36. Momentum of a Proton with respect to an Antiproton.

Follow the link: Checkpoint 52.12.7.

### Subsection 52.15.8 Relativistic Energy

###### Problem 52.15.37. Rest Energy of a Proton.

Follow the link: Checkpoint 52.13.2.

###### Problem 52.15.38. Rest Energy of an Electron.

Follow the link: Checkpoint 52.13.3.

###### Problem 52.15.39. Total Energy of an Ultrarelativistic Proton.

Follow the link: Checkpoint 52.13.4.

###### Problem 52.15.40. Rest Energy of a Pion.

Follow the link: Checkpoint 52.13.5.

###### Problem 52.15.41. Relativistic Kinetic Energy of a Proton.

Follow the link: Checkpoint 52.13.6.

###### Problem 52.15.42. Relativistic Kinetic Energy of an Electron.

Follow the link: Checkpoint 52.13.7.

###### Problem 52.15.43. Energy Released in Collision of a Proton and an Antoproton.

Follow the link: Checkpoint 52.13.8.

###### Problem 52.15.44. Momentum Conservation in a Nuclear Reaction.

Follow the link: Checkpoint 52.13.9.

###### Problem 52.15.45. Energy and Momentum Conservation in a Pion Decay.

Follow the link: Checkpoint 52.13.10.

### Subsection 52.15.9 Miscellaneous Problems

###### Problem 52.15.46. Relativistic Form of \(\vec F = m \vec a\text{.}\).

Consider the relativistic form of \(\vec F = m \vec a\) given in the chapter. Show that if force \(\vec F\) is parallel to the velocity \(\vec u\text{,}\) then this equation becomes

Let \(\vec u = u \hat i\text{.}\)

See solution.

Let \(\vec u = u \hat i\text{.}\) Since \(\vec F \parallel \vec u\text{,}\) \(\vec F = F \hat i\text{.}\) Then

Therefore,

###### Problem 52.15.47. Fizeau Experiment to Velocity Addition Rule.

In 1851 Hippolyte Fizeau carried out an experiment to measure the speed of light in moving water. When water is stationary the speed is

where \(n\) is the refractive index of water. Fizeau found that if water was flowing with speed \(v\) in the same direction as the direction of light his data for the speed of light would fit with the following formula. \(u = \frac{c}{n} + v - \frac{v}{n^2}.\) This formula is an approximate formula for an exact formula that you can derive based on velocity transformation law you have studied in this chapter.

(a) Show that the exact expression for the speed of light will be

(b) Expand the expression in part (a) for \(v/c \ll 1\) keeping only the linear term in \(v/c\) and show that you get the approximate formula agrees with Fizeau's experiment.

Use Maclauren series in \(v/nc\text{.}\)

Already given.

(a) We just add thwe velocity \(v\) of water to that of the velocity of light in the medium \(c/n\text{.}\) This gives

(b) Expand the denominator of the formula upto linear term in \(v/c\text{.}\)

Combining this with the rest of the terms we get the following when we drop quadratic term in \(v/c\text{.}\)

###### Problem 52.15.48. Hafele and Keating Experiment to Test Time Dilation.

Cesium atomic clocks were flown around the world, once eastward and once westward, and their times were then compared with a reference clock at the U.S. Naval Observatory. The time difference in the clocks of the Eastward with Wesward flights cancels out the time dilation due to gravity so that we can obtain the time difference due to special relativity even at the typical aircraft speeds.

Let \(R\) be the radius of Earth, \(\omega\) the angular rotation speed of Earth, \(u\) the speed of flight with respect to the surface of Earth. Below we will calculate the difference in times due to motion if ground speed of the planes is \(u\text{.}\) We will assume that a clock based at the center of Earth is an inertial clock and the clocks at Earth's surface and in the planes move with uniform speed relative the clock at rest at the center of Earth.

(a) Let a clock be at rest in a frame \(S'\) wich is moving with speed \(v\) with respect to another frame \(S\text{.}\) The, show that if \(v \ll c\text{,}\)

Or,

(b) Use the simplified formula for the time dilation to find the times elapsed \(t_s\) for the clock at the surface of Earth if the time elapsed in the clock at the center of the Earth is \(t_0\text{.}\)

(c) Find time \(t_a\) elapsed in a clock in the air plane if the time elapsed in the clock at the center of the Earth is \(t_0\text{.}\) Assume the airplane is going towards East with speed given by \(R\omega + u\text{,}\) which amounts to ignoring the altitude of the plane. Also, note the change in the formula if the plane is going towards West.

(d) Find the formula for difference in the times in planes going East and going West.

(e) Use your relation to find the time difference between the Earth-based clock and the flying clock for a 48-hour trip as observed in the Earth-based clock. Use speed of the flight to be \(u = 232 \text{ m/s}\) and ignore the altitude of the plane compared to the radius of Earth.

Follow the instructions in the problem.

(e) \(4.13\times 10^{-10}\text{ s}\text{.}\)

(a) From the observation that clock in its rest frame goes slower by \(\gamma\) we have

where we can expand \(\gamma\) in powers of \((v/c)^2\text{.}\)

Dropping higher powers gives the answer. Similarly,

(b) We place a clock at surface of Earth. So, \(t_s\) is the time in the \(S'\) frame of (a). Hence, with \(v = \omega R\) since moving in a circle of radius \(R\) with angular speed \(\omega\text{.}\)

(c) Similar to (b), with speed \(v=\omega R + u\text{,}\) we get time in the easward moving plan

For flight in the opposite direction \(u\) will change sign with respec to \(\omega R\text{.}\) Therefore,

(d) The difference will be

(e) Now, we use the numerical values. From \(\omega= 2\pi/24\text{hr}\) and \(R=6.37\text{ km}\text{,}\) we get \(t_0\) from \(t_s= 48\text{ hr}\text{.}\) First we have

Then, we calculate

Now, we evaluate the difference