Question

you are planning a round-trip in which you go in a spacecraft and travel at a constant velocity for exactly 8 months 1-way, as measured by a clock on board the spacecraft, and then return home (without stopping) at the same speed. the overall round-trip (without stoping) would have felt like 12 years in total for the people on earth. according to special relativity, how fast must you travel? (hint: six months is the time one way from the astronaut's perspective, think about the time round trip. 12 years is the time earthians felt the round-trip took)

Answer 1

To make a round-trip feel like 12 years on Earth while the astronaut experiences 8 months each way, the spacecraft must travel at approximately 99.49% the speed of light.

Step-by-step explanation:

The student is asking about the concept of time dilation in special relativity, which arises when comparing the passage of time between two observers moving relative to each other at high speeds. To determine the velocity the astronaut must travel to make the round-trip from the perspective of the people on Earth feel like 12 years, whilst the astronaut experiences only 8 months each way (1-way), we can use the Lorentz factor (gamma, γ).

Firstly, let us calculate the total time experienced by the astronaut, which is 8 months for the round trip. Since 8 months is ⅔ of a year, it amounts to 1.3333 years. The time dilation factor, γ, tells us how much the astronaut's time is dilated from the perspective of the Earth observers. So using γ = 12 years (Earth time) / 1.3333 years (Astronaut's time) gives us γ ≈ 9.

Now, we can find the velocity using the relation γ = 1 / sqrt(1 - v^2/c^2), where v is the velocity of the spacecraft and c is the speed of light. Solving for v gives us v ≈ 0.9949c. Therefore, the spacecraft must travel at approximately 99.49% the speed of light for the round trip to feel like 12 years to people on Earth while it feels like only 8 months for the astronaut each way.