Calculation of the velocity of a Spaceship moving Relativistically

[Owen-]

Homework Statement

A spaceship travel from Earth $\alpha$-Centauri (4.3 light years away) at a constant velocity. If the time elapsed onboard during the journey is 4.e years, what is the speed of the spaceship?

The spaceship is powered by a drive which works by ejecting protons behind the ship at a velocity relative to the ship of c/2. At what velocity would a person on Earth observer the protons to be travelling, just as the spaceship reached constant velocity as it passed Earth on its way to $\alpha$-Centauri

Homework Equations

$\gamma=(1-\beta^2)^{-1/2}$
$\beta = \frac{v}{c}$
$x'=\gamma(x-vt)$
$t'=\gamma(t-\frac{x \times v^2}{c^2})$

The Attempt at a Solution

$x'=\gamma(4.3-vt)$
$4.3=\gamma(t-\frac{x \times v^2}{c^2})$

Well essentially I couldn't see any way to cancel things with simultaneous equations, and without one other variable known I assumed it to be impossible, unless they want the answer to be in terms of something. Is this correct? Any help, even a hint in the right direction would be much appreciated. :)

Thanks in advance,
Owen.

 

[Owen-]

Any way to fix the typo in the title? :S

 

[vela]

Homework Statement

A spaceship travel from Earth $\alpha$-Centauri (4.3 light years away) at a constant velocity. If the time elapsed onboard during the journey is 4.e years, what is the speed of the spaceship?

What does 4.e mean?

The spaceship is powered by a drive which works by ejecting protons behind the ship at a velocity relative to the ship of c/2. At what velocity would a person on Earth observer the protons to be travelling, just as the spaceship reached constant velocity as it passed Earth on its way to $\alpha$-Centauri

Homework Equations

$\gamma=(1-\beta^2)^{-1/2}$
$\beta = \frac{v}{c}$
$x'=\gamma(x-vt)$
$t'=\gamma(t-\frac{x \times v^2}{c^2})$

The Attempt at a Solution

$x'=\gamma(4.3-vt)$
$4.3=\gamma(t-\frac{x \times v^2}{c^2})$

Well essentially I couldn't see any way to cancel things with simultaneous equations, and without one other variable known I assumed it to be impossible, unless they want the answer to be in terms of something. Is this correct? Any help, even a hint in the right direction would be much appreciated. :)

Any way to fix the typo in the title? :S

What typo?

 

[Chestermiller]

You can do this problem using length contraction and time dilation, but the simplest and most foolproof method of doing it is to apply the LT directly.
Let S (coordinates x,t) represent the rest frame of the Earth and α-centauri, and let S' (coordinates x',t') represent the rest frame of the spaceship. There are two events that are relevant:
I. Spaceship leaves earth
II. Spaceship arrives at α-centauri.

The coordinates in the S and S' frames of reference for the two events are as follows:

I. x = 0, t = 0, x' = 0, t' = 0

II. x = 4.3c, t = ?, x' = 0, t' = 4.

where x' = 0 corresponds to the location of the spaceship in its own rest frame of reference. For this problem, it is more convenient to work with the inverse LT:
$$x=\gamma (x'+vt')$$
$$t=\gamma (t'+\frac{vx'}{c^2})$$

You can use the first of these equations to solve part 1 for the velocity of the spaceship by substituting in the parameters for event 2 (event 1 already satisfies these equations identically).

 

[Owen-]

What does 4.e mean?

sorry it should read 4.3. That might have caused some confusion.

What typo?

I'm just being stupid, there is none - Relativistically looked incorrect to me on 2nd glance :P

Chestermiller, thanks! I think I can do this now, wasn't considering needing the LT at the two distinct points - its a lot easier now that I'm looking at it properly! :)