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Chapter 35: Problem 31

Suppose NASA discovers a planet just like Earth orbiting a star just like the Sun. This planet is 35 light-years away from our Solar System. NASA quickly plans to send astronauts to this planet, but with the condition that the astronauts would not age more than 25 years during this journey. a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during this journey? b) According to the astronauts, what will be the distance of their trip?

Short Answer

Expert verified
The spaceship's speed is approximately 2 * sqrt(6) * 10^8 m/s / 7 in the Earth's reference frame. b) What is the distance of the trip according to the astronauts? The distance of the trip according to the astronauts is approximately (35 light-years) * (5/7).

Step by step solution

01

a) Finding the spaceship's speed

We will use the time dilation formula for this step. The time dilation formula can be given as: Dilated time = Proper time / sqrt(1 - v^2/c^2) where - Dilated time is the time experienced in the Earth's frame (35 years), - Proper time is the time experienced by the astronauts (25 years), - v is the spaceship's speed we need to determine, - c is the speed of light (approx. 3.0 x 10^8 m/s). Let's rearrange the equation to solve for v: v = c * sqrt(1 - (Proper time / Dilated time)^2) Substituting the given values and calculating the speed, we get: v = (3.0 x 10^8 m/s) * sqrt(1 - (25 years / 35 years)^2)
02

a) Calculating the spaceship's speed

Calculate the spaceship's speed: v ≈ (3.0 x 10^8 m/s) * sqrt(1 - (5/7)^2) v ≈ (3.0 x 10^8 m/s) * sqrt(1 - 25/49) v ≈ (3.0 x 10^8 m/s) * sqrt(24/49) v ≈ (3.0 x 10^8 m/s) * (2 * sqrt(6) / 7) v ≈ 2 * sqrt(6) * 10^8 m/s / 7 So, the spaceship must travel at approximately 2 * sqrt(6) * 10^8 m/s / 7 in the Earth's reference frame.
03

b) Finding the distance according to the astronauts

We will use the length contraction formula for this step. The formula can be given as: Contracted length = Proper length * sqrt(1 - v^2/c^2) where - Contracted length is the distance according to the astronauts, - Proper length is the distance measured in the Earth's frame (35 light-years), - v is the calculated spaceship's speed, and - c is the speed of light. Let's rearrange the equation to insert the values: Contracted length = (35 light-years) * sqrt(1 - (2 * sqrt(6) * 10^8 m/s / 7)^2 / (3.0 x 10^8 m/s)^2)
04

b) Calculating the distance according to the astronauts

Calculate the contracted length: Contracted length ≈ (35 light-years) * sqrt(1 - (24/49)) Contracted length ≈ (35 light-years) * sqrt(25/49) Contracted length ≈ (35 light-years) * (5/7) So, according to the astronauts, the distance of their trip would be approximately (35 light-years) * (5/7).

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