Question

-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 not age more than 25 years during the journey a) At what speed must the spaceship travel, in Earth's reference frame, so that the astronauts age 25 years during their journey? b) According to the astronauts, what will be the distance of their trip?

Short answer

Answer: The spaceship must travel at approximately 0.912 times the speed of light (≈ 0.912 * 3*10^8 m/s), and according to the astronauts' frame, the trip will be approximately 20.125 light-years long.

Step-by-step solution

Determine time dilation formula

We can use the time dilation formula, which relates the time experienced by a moving observer (the astronauts, t') and a stationary observer (Earth, t) to find the required speed. The formula is: t' = t * sqrt(1 - (v^2 / c^2)) where t is the time taken by stationary observer, t' is the time taken by moving observer, v is the spaceship's speed, and c is the speed of light.

Substitute given values

The Earth's time (t) is 35 years because the planet is 35 light-years away, and the astronauts' time (t') is 25 years because they should not age more than 25 years. The speed of light (c) is 3*10^8 m/s. We have: 25 = 35 * sqrt(1 - (v^2 / (3*10^8)^2))

Solve for v

Solving for the spaceship's speed (v), we first isolate the term with v^2: (25 / 35)^2 = 1 - (v^2 / (3*10^8)^2) Now, we can find v^2: v^2 = (3*10^8)^2 * (1 - (25 / 35)^2) Take the square root of both sides to find v: v = 3*10^8 * sqrt(1 - (25 / 35)^2)

Calculate v

Computing the value of v, we get: v ≈ 0.912 * 3*10^8 m/s So, the spaceship must travel at roughly 0.912 times the speed of light to ensure that the astronauts will age only 25 years during the journey. b) Find the distance traveled according to the astronauts' frame

Determine length contraction formula

We can use the length contraction formula, which relates the distance observed by a moving observer (the astronauts, L') and a stationary observer (Earth, L). The formula is: L' = L * sqrt(1 - (v^2 / c^2))

Substitute given values

We know the distance observed by Earth (L) is 35 light-years, and we already found the spaceship's speed (v). Converting light-years to meters and then plugging in these values, we get: L' = 35 * (9.461*10^15) * sqrt(1 - (0.912 * 3*10^8)^2 / (3*10^8)^2)

Calculate L'

Computing the value of L', we get: L' ≈ 0.575 * 35 * (9.461*10^15) meters Finally, we convert L' back to light-years: L' ≈ 0.575 * 35 light-years ≈ 20.125 light-years According to the astronauts' frame, their trip will be approximately 20.125 light-years long.