You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. Have you aged more or less than your friends at home?
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35. Special Relativity
Inertial Reference Frames
Problem 63c
Textbook Question
A rocket is fired from the earth to the moon at a speed of 0.990c. Let two events be 'rocket leaves earth' and 'rocket hits moon.' Repeat your calculations of part a if the rocket is replaced with a laser beam.

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Step 1: Understand the problem. The rocket is traveling at a speed of 0.990c (where c is the speed of light) from Earth to the Moon. The two events are 'rocket leaves Earth' and 'rocket hits Moon.' The problem asks to repeat the calculations for a laser beam, which travels at the speed of light (c). This involves concepts of special relativity, particularly time dilation and simultaneity.
Step 2: Define the reference frames. There are two reference frames: the Earth-Moon frame (stationary relative to the rocket and laser beam) and the rocket/laser beam frame (moving relative to the Earth-Moon frame). For the laser beam, its speed is c, and it does not experience time dilation since it travels at the speed of light.
Step 3: Calculate the time taken for the rocket to travel from Earth to the Moon in the Earth-Moon frame. Use the formula for time: , where d is the distance between Earth and Moon, and v is the velocity of the rocket (0.990c). For the laser beam, its velocity is c, so the time taken in the Earth-Moon frame is .
Step 4: Consider the relativistic effects for the rocket. Time dilation occurs because the rocket is moving at a relativistic speed. The proper time experienced by the rocket is given by the formula: . For the laser beam, no time dilation occurs because it travels at the speed of light.
Step 5: Compare the results. The rocket's travel time in the Earth-Moon frame is longer than the laser beam's travel time due to its slower speed. In the rocket's frame, the proper time is shorter due to time dilation. For the laser beam, the travel time in the Earth-Moon frame is simply , and no proper time is experienced by the beam itself.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relativity of Simultaneity
The relativity of simultaneity is a concept from Einstein's theory of relativity that states that two events that are simultaneous in one frame of reference may not be simultaneous in another. This is particularly important when considering high-speed travel, such as a rocket moving at a significant fraction of the speed of light (0.990c), as observers in different frames will perceive the timing of events differently.
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Intro to Relative Motion (Relative Velocity)
Time Dilation
Time dilation is a phenomenon predicted by the theory of relativity, where time passes at different rates for observers in different frames of reference, especially at high velocities. For an observer on Earth, the time taken for the rocket to travel to the moon will differ from the time experienced by the rocket itself, which will be less due to its high speed approaching the speed of light.
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Time Dilation
Speed of Light as a Constant
The speed of light in a vacuum is a fundamental constant of nature, approximately 299,792 kilometers per second (or about 186,282 miles per second). According to relativity, no object with mass can reach or exceed this speed, and light itself serves as a universal speed limit. This concept is crucial when comparing the travel times of a rocket and a laser beam, as the latter travels at the speed of light, while the former does not.
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Phase Constant of a Wave Function
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