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Chapter 26: Problem 9

A spaceship travels at constant velocity from Earth to a point 710 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is $0.9999 c .$ A passenger is 20 yr old when departing from Earth. (a) How old is the passenger when the ship reaches its destination, as measured by the ship's clock? (b) If the spaceship sends a radio signal back to Earth as soon as it reaches its destination, in what year, by Earth's calendar, does the signal reach Earth? The spaceship left Earth in the year 2000.

Short Answer

Expert verified
Answer: The passenger will be approximately 51.812 years old when the spaceship reaches its destination. The radio signal sent from the spaceship upon arrival at its destination will be received back on Earth around the year 2710.

Step by step solution

01

Calculate the time experienced by Earth (t)

First, we need to find the time experienced by Earth (denoted by \(t\)) during the spaceship's journey. Since the distance is 710 light-years and the ship's speed is 0.9999c, we can calculate the time as follows: $$t = \frac{distance}{speed} = \frac{710}{0.9999}$$ Calculating this value, we get: $$t \approx 710.071$$ So, Earth experiences about 710.071 years during the spaceship's journey.
02

Calculate time dilation (t')

Now, we use the time dilation formula to find the time experienced by the passenger aboard the spaceship (denoted by \(t'\)): $$t'= t\sqrt{1-\frac{v^2}{c^2}}$$ Plugging in the values for \(t\) and \(v\), we have: $$t' = 710.071\sqrt{1-\frac{(0.9999c)^2}{c^2}}$$ Evaluating this expression, we get: $$t' \approx 31.812$$ So, the passenger aboard the spaceship experiences about 31.812 years during the journey.
03

Determine the age of the passenger

To find the age of the passenger when the spaceship reaches its destination, we add the time experienced by the passenger (31.812 years) to their initial age (20 years): $$age_{final} = age_{initial} + t' = 20 + 31.812$$ Calculating this, we get: $$age_{final} \approx 51.812$$ The passenger is about 51.812 years old when the spaceship reaches its destination.
04

Determine the year when a radio signal reaches Earth

Since the radio signal travels at the speed of light, it will take the same amount of time experienced by Earth (710.071 years) to reach Earth after being sent from the spaceship. The spaceship left Earth in the year 2000, so to find the year when the signal reaches Earth, we add the time experienced by Earth to the year 2000: $$year_{signal} = year_{departure} + t = 2000 + 710.071$$ Calculating this, we get: $$year_{signal} \approx 2710.071$$ So, the radio signal sent from the spaceship upon arrival at its destination reaches Earth around the year 2710.

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