How Do You Calculate Relative Velocity in Opposing Spaceships?

[Infernorage]

Homework Statement

Spaceship A and spaceship B are each a proper length 100.0 m and travel at constant velocities. The spaceships pass in outer space while traveling in opposite directions. The pilot of A observes that 2 microseconds elapse while spaceship B passes by the cockpit window, heading in the opposite direction.
(a) What is the magnitude of the relative velocity of the ships?

The Attempt at a Solution

So, I did 100m/(2x10^-6 seconds) and I got a velocity of 5x10^7m/s. I don't know what to do after that though. It doesn't seem like there is enough information is given to use the relative velocity equation, but the answer can't just be that simple, right?

Can you guys tell me what I need to do next or if that answer is correct? Thanks in advance.

 

[D H]

The answer is not that simple.

Hint: What is 5x10⁷ m/s as a fraction of the speed of light?

 

[Infernorage]

The answer is not that simple.

Hint: What is 5x10⁷ m/s as a fraction of the speed of light?

Oh okay, I knew that couldn't be right. Well, 5x10⁷ m/s is equal to about 0.167C. So, am I supposed to find the relative change in time and the length of ship A that ship B sees in order to calculate the speed at which ship B see A pass by?

 

[Infernorage]

So should I go the route I said above?

 

[D H]

How much relativity do you know? This is obviously a relativity problem.

 

[Infernorage]

How much relativity do you know? This is obviously a relativity problem.

I haven't been learning relativity for very long, which is why I wanted to know if the direction I was going was correct; and if it wasn't correct, I was hoping for some guidance.

 

[D H]

You already know the time: it is 2 microseconds. How long is spaceship B from the perspective of the pilot of spaceship A?

 

[Infernorage]

You already know the time: it is 2 microseconds. How long is spaceship B from the perspective of the pilot of spaceship A?

Okay, so you would use the length contraction equation and use 100m for proper length, and .167C for the speed?

 

[D H]

You don't know what the velocity is yet.

 

[diazona]

Well, actually the speed that you figured out (.167c) is not exactly correct. It's a small fraction of the speed of light so I'd expect it to be close, but the problem probably wants you to find an exact solution.

Think about it this way: you know that moving objects appear to be length-contracted, right? This means that, although spaceship B would measure itself to be 100m long, spaceship A would measure its length to be less than that. Just how much less, you don't know, because you're not given the relative speed of the two spaceships. But you can write a formula for it.

Pick a variable, let's say x, to represent the length of spaceship B from the perspective of spaceship A (i.e. the contracted length). Now can you write the formula that relates x to the uncontracted length (100m) of the spaceship? It will involve the relative speed of the two ships, which you also don't know - so use a variable for that too, let's say v.

Now, imagine things from the perspective of spaceship A. The pilot sees spaceship B pass by him in a time of 2 microseconds, and he observes that the length of spaceship B is x. If you see an object of length x (as measured by you) take a time t (as measured by you) to pass by you, what is its speed relative to you?

From the two paragraphs above, you should have gotten two different formulas involving the relative speed of the two ships and the unknown contracted length x. Given a system of two equations and two unknown variables, you should be able to find the value of the variable you want.

 

[Infernorage]

I figured it out. Thank you very much for the help.