# Velocity of relativistic spaceship

1. May 18, 2014

### Rct33

1. The problem statement, all variables and given/known data
A spaceship travels from Earth to a star that is 6 light years away. The spaceship takes 2.5 years to reach the star in its frame. Calculate the velocity of the spaceship.

2. Relevant equations
$x=\frac{x_0}{γ}$, $t=γt_0$

3. The attempt at a solution
I guess I have to relate the two equations to work out velocity somehow. Previous attempts where I considered the distance, $6ly$ divided by the velocity of the spaceship was equal to the time it takes to travel to the star as seen on Earth. I then substituted $t$ for $γt_0$ where $t_0=2.5y$ and rearranged to find $v$, but this was unsuccessful. I don't have any other ideas to try so hints would be appreciated.

2. May 18, 2014

### Staff: Mentor

Your approach looks good. Please show your work so we can see where the calculations went wrong.

3. May 18, 2014

### Rct33

$\frac{6}{v}=t=2.5γ$ where the velocity is a fraction of c, 6 is in light years and t, 2.5 are in years.

Implies:

$\frac{6}{2.5}=\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}$

$∴v=\frac{12c}{\sqrt{25c^2 + 144}}=2.4$ solved with wolfram because tired

Can't understand why I get 2.4c as an answer?

4. May 18, 2014

### Staff: Mentor

That should be: $\frac{6c}{v}=t=2.5γ$

Chet

5. May 18, 2014

### Rct33

Cheers

6. May 18, 2014

### Staff: Mentor

$$\frac{6}{2.5}=\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}$$
This has a solution with v smaller than 1.

$$∴v=\frac{12c}{\sqrt{25c^2 + 144}}=2.4$$ Don't use c here (or plug in 1), as you worked with years=speed of light = 1 anyway.

A proper calculation with units would not have this issue...