Relativity problem - spaceship passing transmitting station

[w3390]

Homework Statement

A spaceship of proper length Lp = 450 m moves past a transmitting station at a speed of 0.61c. (The transmitting station broadcasts signals at the speed of light) A clock is attached to the nose of the spaceship and a second clock is attached to the transmitting station. The instant that the nose of the spaceship passes the transmitter, clocks at the transmitter and in the nose of the spaceship are set to zero. The instant that the tail of the ship passes the transmitter a signal is sent and subsequently detected by the receiver in the nose of the ship.

When, according to the clocks attached to the nose of the spaceship, is the signal received?

Homework Equations

TIME DILATION: \Deltat=\gamma*\Deltat(proper)
LENGTH CONTRACTION: L=L(proper)/\gamma
\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

The Attempt at a Solution

The question before the one included in part a asked when, according to the clock at the nose of the ship, was the signal sent? I got the answer for that by simply dividing the length of the spaceship [450m] by .61c [ 61% the speed of light]. This gave me 2.5 microseconds which is correct. However, when I tried to answer the question posed in part a, my answer is wrong. My method was to figure out the time dilation using the 2.5 microseconds and multiplying it by gamma. This gave me 3.1 microseconds. Am I correct in my logic in adding the 3.1 microsecond time dilation to the 2.5 microsecond proper time to get 5.6 microseconds for the time to receive the signal?

Any help is much appreciated.

 

[fantispug]

Let's start from your first answer: according to the nose of the spaceship the transmitting station emits a signal after 2.5 microseconds.

I don't understand what you are doing from here; you apply the time dilation formula to this quantity; in what frame is 2.5 microseconds a proper time interval for something relevant?

Let's just think: from the perspective of the nose how fast does the signal the station emit appear to be travelling. How far, in the nose frame, is the transmitter from the nose? Using these two figures how much longer does it take for the signal to get to the nose?

 

[w3390]

Let's just think: from the perspective of the nose how fast does the signal the station emit appear to be travelling.

I am confused by what you said above because the problem states that the signal is traveling at the speed of light. Therefore, it should appear the same in all reference frames. Do I need to do some sort of velocity transformation using the equation:

u'=\frac{u-v}{1-\frac{vu}{c^2}}

However, it seems as though I would have to use some sort of time dilation formula because the next question asks the same thing but from the frame of the transmitter.

 

[w3390]

Okay, disregard my logic in the above post. Since posting I have thought about the problem and have come to the conclusion that it perhaps is just a problem of comparing the velocity of the transmission (c) to the velocity of the spaceship (.61c). I set up two equations with distance=velocity*time and set the equations equal to each other for equal distance. I worked through the algebra and got the answer 3.8 microseconds for the time when the signal is received. This answer seems reasonable compared to the answer of 2.5 microseconds for the time when the signal was sent. Is this logic correct and does it yield the correct answer?

 

[w3390]

So it seems that all the formulas I have thought to have been correct are just overcomplicating the problem. It seems like it should just be a simple x=v*t problem but I do not know what x is. I really need help understanding this problem conceptually.