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Chapter 35: Problem 73

Two stationary space stations are separated by a distance of \(100 .\) light- years, as measured by someone on one of the space stations. A spaceship traveling at \(0.950 c\) relative to the space stations passes by one of the space stations heading directly toward the other one. How long will it take to reach the other space station, as measured by someone on the spaceship? How much time will have passed for a traveler on the spaceship as it travels from one space station to the other, as measured by someone on one of the space stations? Round the answers to the nearest year.

Short Answer

Expert verified
Answer: It will take approximately 105 years for the spaceship to reach the other space station, as measured by someone on the spaceship. For a traveler on the spaceship, 337 years will have passed as measured by someone on one of the space stations during the journey.

Step by step solution

01

Identify the given values

We're given: - Distance between the space stations (D): 100 light-years - Speed of the spaceship (v): 0.950 c
02

Calculate the time it takes to reach the other space station, as measured by someone on the spaceship

We can use the formula for time as: \(time = \frac{distance}{speed}\) which can be rewritten as: \(t_{spaceship} = \frac{D}{v}\) Then, \(t_{spaceship} = \frac{100}{0.950 c}\) \(t_{spaceship} = \frac{100}{0.950} \,\text{years}\) \(t_{spaceship} \approx 105 \,\text{years}\)
03

Calculate time dilation factor

We can find the time dilation factor using the formula: \(\gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}\) where \(c\) is the speed of light. \(\gamma = \frac{1}{\sqrt{1- \frac{(0.950c)^2}{c^2}}}\) \(\gamma = \frac{1}{\sqrt{1- 0.9025}}\) \(\gamma \approx 3.203\)
04

Calculate the time that passes for a traveler on the spaceship, as measured by someone on the space station

We can use the time dilation factor to find the time that passes for a traveler on the spaceship (\(t_{station}\)) as: \(t_{station} = \gamma \times t_{spaceship}\) \(t_{station} = 3.203 \times 105\,\text{years}\) \(t_{station} \approx 337\,\text{years}\) So, it will take approximately 105 years for the spaceship to reach the other space station, as measured by someone on the spaceship, and 337 years will have passed for a traveler on the spaceship as it travels from one space station to the other, as measured by someone on one of the space stations.

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Most popular questions from this chapter

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.650 c .\) This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of \(0.914 c\) to establish a relative speed of \(0.650 c\) with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

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