Finding relative velocity when there are multiple unknowns

[fifel85]

Homework Statement

Two spaceships, each measuring 300 m in its own rest frame, pass by each other traveling in opposite directions. Instruments on board spaceship A determine that the front of spaceship A requires 1 microsecond to traverse the full length of B. What is the relative velocity of the two spaceships (in units of the speed of light)?

Homework Equations

In this course, we are not using gamma in the Lorentz factor...just sqrt(1-v^2/c^2)

The Attempt at a Solution

I know that this is a relativity problem involving length contraction. There are two unknowns - relative velocity (v) and the contracted length of Spaceship B. Using the Lorentz factor, I know that the contracted length (L) of spaceship B is: L=300m(sqrt(1-v^2/c^2)). Because velocity is distance/time, I also know that v = L/1microsecond.
I substituted L=300m(sqrt(1-v^2/c^2)) for L in this second equation, to give:

v = ((300m(sqrt(1-v^2/c^2)/1 microsecond)

Unfortunately, here I end up with a messy equation with v on both sides, and I'm having trouble solving for v. Am I just not doing the algebra right? Is there a way to simplify this?

 

[BruceW]

You've done good so far. And yes, it is a bit messy, but if you rearrange, then you can solve for v.

Or there is a short-cut, which you wouldn't ordinarily be able to use, but it is possible in this case because of the numbers which they have given you. If you look at the numbers, you've got 300m / 1 microsecond so what constant could you divide this by to get a nice number?

 

[fifel85]

Oh, ok, thanks! 300m/1 microsecond is the speed of light (c), so I can just separate that out...so v = c (sqrt(1-v^2/c^2). Solving for v, I get 2.12*10^8 m/s relative velocity, or about 70% the speed of light. ? Thanks so much!

 

[BruceW]

yep, that's right :) Often in exams, they do use convenient numbers so that you can use short-cuts such as this, but you can also practise rearranging the equation, in case they give you a question with not-convenient numbers.