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Chapter 26: Problem 5

An astronaut wears a new Rolex watch on a journey at a speed of $2.0 \times 10^{8} \mathrm{m} / \mathrm{s}$ with respect to Earth. According to mission control in Houston, the trip lasts \(12.0 \mathrm{h}\). How long is the trip as measured on the Rolex?

Short Answer

Expert verified
Answer: The trip's duration as measured on the astronaut's Rolex watch is approximately 57,915 seconds or approximately 16.1 hours.

Step by step solution

01

Identify the given values

We are given the speed of the astronaut's journey, \(v = 2.0 \times 10^8 \, \mathrm{m/s}\), with respect to Earth, and the trip lasts a time duration \(t_{earth} = 12.0 \, \mathrm{h}\) according to mission control in Houston. Our task is to determine the time duration as measured by the astronaut's Rolex watch, \(t_{astronaut}\).
02

Convert the Earth time duration to seconds

Before proceeding with the time dilation formula, we need to convert the given Earth time duration from hours to seconds. This will make our calculations consistent in terms of units: \[t_{earth} = 12.0 \, \mathrm{h} \times \frac{3600 \, \mathrm{s}}{1 \, \mathrm{h}} = 43200 \, \mathrm{s}\]
03

Apply the time dilation formula

Next, we will use the time dilation formula derived from the Lorentz transformation to relate the time experienced by the astronaut to the time experienced by mission control on Earth. The formula is as follows: \[t_{astronaut} = \frac{t_{earth}}{\sqrt{1 - \frac{v^2}{c^2}}}\] In this equation, \(v\) is the velocity of the astronaut with respect to Earth, \(t_{earth}\) is the time duration experienced on Earth, and \(c\) represents the speed of light, which is approximately \(3.0 \times 10^8 \, \mathrm{m/s}\).
04

Calculate the time experienced by the astronaut

Now we will plug in the given values and solve for the time experienced by the astronaut, \(t_{astronaut}\): \[t_{astronaut} = \frac{43200 \, \mathrm{s}}{\sqrt{1 - \frac{(2.0 \times 10^8 \, \mathrm{m/s})^2}{{(3.0 \times 10^8 \, \mathrm{m/s})}^2}}}\]
05

Compute the final result

By calculating the value inside the square root and solving the equation above, we can find the value of \(t_{astronaut}\): \[t_{astronaut} \approx \frac{43200 \, \mathrm{s}}{\sqrt{0.5556}} \approx \frac{43200 \, \mathrm{s}}{0.7454} \approx 57915 \, \mathrm{s}\] So, the trip's duration as measured on the Rolex watch is approximately \(57915 \, \mathrm{s}\) or approximately \(16.1 \, \mathrm{h}\).

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Most popular questions from this chapter

The solar energy arriving at the outer edge of Earth's atmosphere from the Sun has intensity \(1.4 \mathrm{kW} / \mathrm{m}^{2}\) (a) How much mass does the Sun lose per day? (b) What percent of the Sun's mass is this?
A spaceship is traveling away from Earth at \(0.87 c .\) The astronauts report home by radio every \(12 \mathrm{h}\) (by their own clocks). At what interval are the reports sent to Earth, according to Earth clocks?
Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at \(0.40 c\) relative to the Earth observer, and ship \(\mathrm{B}\) moves at \(0.50 c\) relative to the same observer. What speed does the captain of ship A report for the speed of ship B?
A spaceship is moving away from Earth with a constant velocity of \(0.80 c\) with respect to Earth. The spaceship and an Earth station synchronize their clocks, setting both to zero, at an instant when the ship is near Earth. By prearrangement, when the clock on Earth reaches a reading of $1.0 \times 10^{4} \mathrm{s}$, the Earth station sends out a light signal to the spaceship. (a) In the frame of reference of the Earth station, how far must the signal travel to reach the spaceship? (b) According to an Earth observer, what is the reading of the clock on Earth when the signal is received?
The fastest vehicle flown by humans on a regular basis is the Space Shuttle which has a mass of about \(1 \times 10^{5} \mathrm{kg}\) and travels at a speed of about \(8 \times 10^{3} \mathrm{m} / \mathrm{s} .\) Find the momentum of the Space Shuttle using relativistic formulae and then again by using nonrelativistic formulae. By what percent do the relativistic and nonrelativistic momenta of the Space Shuttle differ? [Hint: You might want to use one of the approximations in Appendix A.5.]
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