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

[EngageEngage]

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:
$$t' = 1.0*10^-6s... x' = 300m... V = .8c$$

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

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!

 

[EngageEngage]

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

 

[Moetasim]

First of all, great job on part a! Your approach was correct and you got the right answer.

For part b, you can use the fact that the speed of light is the same in all reference frames. This means that the time it takes for the light pulse to travel from the tail to the nose of the spaceship should be the same in both the spaceship's reference frame and Earth's reference frame. In other words, the time interval t' in the spaceship's frame should be equal to the time interval t in Earth's frame.

So, using the equation t = x/c, where x is the distance traveled by the light pulse, we can solve for t in Earth's frame:
t = x/c = 300m/c = 1.0 * 10^-6 s

Now, to use the Lorentz transformations to calculate this time, we can use the equation you provided in your attempt:
t =\frac{t' + \frac{Vx'}{c^{2}}}{\sqrt{1-\frac{V^{2}}{c^{2}}}}

Plugging in the values:
t =\frac{1.0*10^-6s + \frac{(.8c)(300m)}{c^{2}}}{\sqrt{1-\frac{(.8c)^{2}}{c^{2}}}}
= 1.40 * 10^-6s

As you can see, we got the same result using both methods. This shows that the Lorentz transformations are a valid way to calculate time intervals in different reference frames. I hope this helps!