The discussion revolves around calculating the speed required to reach a star 240 light years away within an 85-year human lifetime, incorporating concepts of time dilation and the distinction between different frames of reference. The conversation touches on theoretical implications and mathematical setups related to relativistic travel.
Participants do not reach a consensus on the initial setup of the problem, with some confusion regarding the definitions of time and distance in the context of the scenario. Multiple viewpoints on the interpretation of the problem and the application of time dilation remain present.
There are unresolved assumptions regarding the initial problem statement and the implications of relativistic effects on time measurement. The discussion reflects varying interpretations of the relationship between time and distance in the context of special relativity.
MermaidWonders said:How fast would you have to go to reach a star 240 light years away in an 85-year human lifetime?
Here, I know that I'm supposed to find $v$, but I'm having a hard time setting up my equation(s) in order to reach the final answer. :(
There is a wrinkle here. If we are talking about an observer on Earth then as tkhunny points out there must be some kind of misprint... It can't be done.MermaidWonders said:How fast would you have to go to reach a star 240 light years away in an 85-year human lifetime?
Here, I know that I'm supposed to find $v$, but I'm having a hard time setting up my equation(s) in order to reach the final answer. :(
tkhunny said:Let's start with a question. You do know that you can go 85 light years in 85 years at 1x Speed of Light, right?
topsquark said:There is a wrinkle here. If we are talking about an observer on Earth then as tkhunny points out there must be some kind of misprint... It can't be done.
If you are talking about an observer on the flight we can do it. What do you know about time dilation? (See the section "Simple Inference of Velocity Time Dilation.")
Can you finish from here? If not just let us know.
-Dan
So what we have here isMermaidWonders said:I know that the time measured in the frame in which the clock is at rest is called the proper time, and so a moving clock runs slower (hence dilated).
topsquark said:So what we have here is
[math]\Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}[/math]
where [math]\Delta t'[/math] is measured by the observer's clock and [math]\Delta t[/math] is the time in the moving frame, aka "proper time."
So one possibility is if we wish the moving frame to experience 85 years and the observer's frame to be 240 years, then [math]\Delta t' = 240[/math] and [math]\Delta t = 85[/math]. Solve for v.
-Dan
topsquark said:You were talking about times in your original post, not distances.
-Dan
MermaidWonders said:What do you mean? It says "240 light years away"... I'm confused.
I like Serena said:Indeed, the distance traveled is $v\Delta t' = 240\text{ lightyear}$.
So we have $\Delta t' = \frac{240\text{ lightyear}}{v}$ and $\Delta t = 85\text{ year}$.
MermaidWonders said:Ah, OK, that makes sense. So is $\Delta t'$ the time measured by the person on board the spaceship, and 85 years measured by a person at rest on Earth?
I like Serena said:It's the other way around.