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Verify that the solution in Example 1 in the discussion is related through time dilation with γ = 30.00 as given.   Example 1

Verify that the solution in Example 1 in the discussion is related through time dilation with γ = 30.00 as given.   Example 1

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Verify that the solution in Example 1 in the discussion is related through time dilation with γ = 30.00 as given.

 

Example 1:

 

RAL PHYSICS 2 Suppose an astronaut, such as the twin discussed in Simultaneity and Time Dilation, travels so fast that y = 30.00. 1. She travels from the Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are the Earth and Alpha Centauri as measured by the astronaut? 2. In terms of c, what is her velocity relative to the Earth? You may neglect the motion of the Earth relative to the Sun. Alpha Centauri Earth (a) Alpha Centauri Strategy: First note that a light year (ly) is a convenient unit of distance on an astronomical scale-it is the distance light travels in a year. For Part 1, note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance Lo, because it is measured by an Earth-bound observer to whom both stars are (approximately) stationary. To the astronaut, the Earth and the Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length L. In Part 2, we are given y, and so we can find v by rearranging the definition of y to express v in terms of c. Given: Lo = 4.300 ly (b) L =? L = 0.1433 ly y = 30.00 Solution for Part 1: L = Lo 4.300 ly L = 30.00 Solution for Part 2: y = 30.00 find "v" in terms of "c" 1 = 1- 900 Y = 30.00 = square both sides → 900 - 1- = 0.9988 %3D 900 Taking the square root, we find; = 0.99944, which rearranged to produce a value for the velocity v 0.99944c Time Dilation The first result of relativity is that time is not the same for all frames of reference. The simplest case for relativity is light
Transcribed Image Text:RAL PHYSICS 2 Suppose an astronaut, such as the twin discussed in Simultaneity and Time Dilation, travels so fast that y = 30.00. 1. She travels from the Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are the Earth and Alpha Centauri as measured by the astronaut? 2. In terms of c, what is her velocity relative to the Earth? You may neglect the motion of the Earth relative to the Sun. Alpha Centauri Earth (a) Alpha Centauri Strategy: First note that a light year (ly) is a convenient unit of distance on an astronomical scale-it is the distance light travels in a year. For Part 1, note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance Lo, because it is measured by an Earth-bound observer to whom both stars are (approximately) stationary. To the astronaut, the Earth and the Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length L. In Part 2, we are given y, and so we can find v by rearranging the definition of y to express v in terms of c. Given: Lo = 4.300 ly (b) L =? L = 0.1433 ly y = 30.00 Solution for Part 1: L = Lo 4.300 ly L = 30.00 Solution for Part 2: y = 30.00 find "v" in terms of "c" 1 = 1- 900 Y = 30.00 = square both sides → 900 - 1- = 0.9988 %3D 900 Taking the square root, we find; = 0.99944, which rearranged to produce a value for the velocity v 0.99944c Time Dilation The first result of relativity is that time is not the same for all frames of reference. The simplest case for relativity is light
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