How Long Does Light Take to Travel in Different Reference Frames?

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Homework Statement

A spaceship has a speed of .8c relative to Earth. In its own reference fram, the length of this spaceship is 300m.
a.) consider a light emiited from the tail of this spaceship. In the reference frame of the spaceship, how long does this pulse take to reach the nose>
b.) In the reference frame of the Earth, how long does this take? Calculate this time directly from the motions of the spaceship and the light pulse; hen recalculate it by applying the Lorentz transformations to the result obtained in (a).

Homework Equations

$$t =\frac{t' + \frac{Vx'}{c^{2}}}{\sqrt{1-\frac{V^{2}}{c^{2}}}}$$

The Attempt at a Solution

I think i have part a figured out.. All i did was divide 300 by c to get $$\Delta t' = 1.0*10^-6s$$

b.) For this part, I cannot figure out how to do it without using the Lorentz transform directly like so:
$$t' = 1.0*10^-6s...<br /> x' = 300m...<br /> V = .8c$$

$$t =\frac{1.0*10^-6s + \frac{.8c(300m)}{c^{2}}}{\sqrt{1-\frac{(.8c)^{2}}{c^{2}}}}<br /> = 1.40 * 10^-6s$$

I cannot do this though without the lorentz transform. We haven't gone over length contraction yet, so I cannot use it to determine the length of the spaceship in Earth's reference frame. If anyone could please help me get started on this I would appreciate it greatly!

 

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I've been trying to manipulate the other lorentz equations ( in specific the ones for x), but i cannot find anything that will work. Once again, If someone could help me out here I would appreciate it