How Does Special Relativity Affect Exam Time and Signal Transmission?

[Lavabug]

Homework Statement

A class of students is taking a 1-hour exam on relativistic kinetics. When the exam starts, both the class and the professor set their stopwatches to zero and the professor leaves with a velocity of 0.86c along the x-axis w/ respect to the class. The professor while traveling sends an electromagnetic signal towards the students such that when it arrives, his stopwatch will display 1 hour.

How much time do the students have to complete the exam, according to their clock?

When did the professor send the signal, according to the students?

Homework Equations

Lorentz transforms.

The Attempt at a Solution

The first question was straight-forward, I used the equation for time dilation and got 0.51hrs for the time the students really have to complete the exam. Time in O' > Time in O so it makes sense.

2nd part of the problem boggles me. I've been playing around with the lorentz contraction for a bit but I keep making a huge mess that doesn't go anywhere. Supposedly I don't need to know the position of the prof to solve this.

 

[Mike Pemulis]

Let's work in the professor's frame. The classroom is traveling 0.86c away from him. He sends the signal at time t0, when the classroom is (.86c)*t0 away. It catches up to the classroom at t = 1 hr. Can you solve for t0? If so, you can transform this to get the time in the classroom frame.

 

[Lavabug]

Don't know how I would solve for t0. Without knowing the actual positions where the signal was sent I'm a bit stuck.

 

[Mike Pemulis]

You're right, we don't know the position where the signal was sent from. But we don't need to know that, because we have two pieces of information:

1. The prof is (.86c)t0 away when he sends the signal.

2. The signal, moving toward the classroom at a relative speed of 0.14c, takes time t = 1 - t0 to make up this distance.

And that gives you t0.

 

[Lavabug]

You're right, we don't know the position where the signal was sent from. But we don't need to know that, because we have two pieces of information:

1. The prof is (.86c)t0 away when he sends the signal.

2. The signal, moving toward the classroom at a relative speed of 0.14c, takes time t = 1 - t0 to make up this distance.

And that gives you t0.

Isn't it moving at c regardless of the reference frame? If that's the case I don't see how its moving at 0.14c.

Is t = the time the signal is received according to the students' clock? So, t0 = 0.49 Hrs was when the prof sent the signal according to their clocks?

 

[ideasrule]

Isn't it moving at c regardless of the reference frame? If that's the case I don't see how its moving at 0.14c.

Is t = the time the signal is received according to the students' clock? So, t0 = 0.49 Hrs was when the prof sent the signal according to their clocks?

Yes, the signal travels at c no matter what. So according to the professor, he's d=0.86c*t0 away when he sends the signal. It must arrive in 1h-t0, so c*(1-t0)=d. Once you know t0, you can use time dilation to figure out when the students think he sent the signal.

 

[Mike Pemulis]

Isn't it moving at c regardless of the reference frame? If that's the case I don't see how its moving at 0.14c.

Okay, I was taking a slight shortcut. The light is moving away from the professor at c, but the classroom is moving away in the same direction at 0.14c. So the professor sees the light moving 0.14c faster than the classroom. If you prefer, we can define the following variables (all in the professor's frame):

t0 = time signal is sent.
x0 = distance between the classroom and prof at t0. So x0 = (0.86c)t0
t1 = time when the signal reaches the classroom. t1 = 1 hour.
x1 = distance between the professor and classroom when the signal reaches the classroom. So x1 = (0.86c)t1.

Write down x1 - x0. Solve for t0. Then time-dilate the result back to the classroom frame.

Edit: Ninja'd -- what ideasrule said.