# PhysicsSpecial Relativity

#### MermaidWonders

##### Active member
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
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
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
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
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
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
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
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
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
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
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.