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Physics Special Relativity

MermaidWonders

Active member
Feb 20, 2018
113
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

Well-known member
MHB Math Helper
Jan 27, 2012
267
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. :(
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

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
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.

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
 

MermaidWonders

Active member
Feb 20, 2018
113
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?
Yes.
 

MermaidWonders

Active member
Feb 20, 2018
113
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
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

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
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).
So what we have here is
\(\displaystyle \Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}\)

where \(\displaystyle \Delta t'\) is measured by the observer's clock and \(\displaystyle \Delta t\) 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 \(\displaystyle \Delta t' = 240\) and \(\displaystyle \Delta t = 85\). Solve for v.

-Dan
 

MermaidWonders

Active member
Feb 20, 2018
113
So what we have here is
\(\displaystyle \Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}\)

where \(\displaystyle \Delta t'\) is measured by the observer's clock and \(\displaystyle \Delta t\) 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 \(\displaystyle \Delta t' = 240\) and \(\displaystyle \Delta t = 85\). Solve for v.

-Dan
But isn't the light year a unit of distance? For instance, the 240 light years in this question would represent a certain distance?
 

MermaidWonders

Active member
Feb 20, 2018
113
You were talking about times in your original post, not distances.

-Dan
What do you mean? It says "240 light years away".... I'm confused.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
What do you mean? It says "240 light years away".... I'm confused.
Indeed, the distance travelled is $v\Delta t' = 240\text{ lightyear}$.
So we have $\Delta t' = \frac{240\text{ lightyear}}{v}$ and $\Delta t = 85\text{ year}$.
 

MermaidWonders

Active member
Feb 20, 2018
113
Indeed, the distance travelled is $v\Delta t' = 240\text{ lightyear}$.
So we have $\Delta t' = \frac{240\text{ lightyear}}{v}$ and $\Delta t = 85\text{ year}$.
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?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
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?
It's the other way around.
 

MermaidWonders

Active member
Feb 20, 2018
113