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Chapter 33: Problem 38

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

Short Answer

Expert verified
For the observer moving at 0.750c from Earth to Sun and at 0.294c from Earth to Sun, event A happens first. However, for the observer moving at 0.750c from the Sun to the Earth, event B happens first.

Step by step solution

01

Rest Frame

First, consider the frame in which Earth and Sun are at rest. Event A is at time \(t=0\) and event B is at \(t=2.45 \mathrm{min}\). Because light takes 8.33 minutes to travel from the Sun to the Earth, event B is observed on Earth at \(t=2.45 \mathrm{min} + 8.33 \mathrm{min}=10.78 \mathrm{min}\). So, event A takes place first.
02

Moving Frame (0.750c)

Now, consider an observer moving from the Earth towards the Sun at 0.750c. We use the Lorentz Transformation to find the time in the moving frame: \(t' = \gamma(t - v \cdot d/c^2)\), where \(v\) is the velocity, \(\gamma = 1/\sqrt{1-v^2/c^2}\), \(d\) is the distance, and \(t\) is the time in the rest frame. Substituting the given values and calculating gives \(t'=2.92 \mathrm{min}\) for event A and \(t'=9.76 \mathrm{min}\) for event B, so event A still happens first.
03

Moving Frame (-0.750c)

Now consider an observer moving from the Sun towards the Earth at 0.750c. Again using the Lorentz Transformation, with \(v=-0.750c\), the times for events A and B are calculated to be \(t'=5.01 \mathrm{min}\) and \(t'=4.56 \mathrm{min}\), respectively. Now, event B happens first.
04

Moving Frame (0.294c)

Lastly, consider an observer moving from the Earth towards the Sun at 0.294c. Using the Lorentz Transformation with \(v=0.294c\), the times for events A and B are calculated to be \(t'=0.43 \mathrm{min}\) for event A and \(t'=8.68 \mathrm{min}\) for event B, so event A happens first.

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Most popular questions from this chapter

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.

By what factor does an object's momentum change if you double its speed when its original speed is (a) \(25 \mathrm{m} / \mathrm{s}\) and (b) \(100 \mathrm{Mm} / \mathrm{s} ?\)

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

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 \(0.80 c\) relative to Earth for 6.67 min according to clocks on Earth, how many atoms will be left at the end of that time?

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