Tangential Velocity, Frames, and the Speed of Light

[Nishmaster]

Hello gurus,

I have a mental experiment that I'm wrangling with.

Assume for a moment that two craft (Ship A and Ship B) are approaching each other at a significant fraction of c. Let us also assume that I, the observer, am in a stationary craft a large distance away rotating with the plane given by the vector of the two approaching craft, keeping one of them (Ship A) stationary in the center of my field of vision.

I have two questions:

1. Since I have no way of measuring the velocity of Ship A in my frame, is it assumed to be zero?

2. What would I measure as the velocity of Ship B?

I'm no mathematician of this level, but by all means don't dumb it down for me, as I'm really trying to pick some of this up.

 

[Mentz114]

You haven't specified the scenario properly because

.. I, the observer, am in a stationary craft ..

has no meaning unless you say in which frame of reference you are stationary. Presumably you mean to take your own ship as the stationary frame ?

Also, I'm puzzled as to why you think you can't measure the velocity of the other ships.

If you can measure one, why not the other ?

 

[JesseM]

1. Since I have no way of measuring the velocity of Ship A in my frame, is it assumed to be zero?

2. What would I measure as the velocity of Ship B?

In what coordinate system are you asking for the velocities of the ships? If you want a coordinate system where every part of your craft is at rest, this would be a rotating coordinate system, and thus a non-inertial one, and there isn't really a single coordinate system which is "the" rest frame of a non-inertial observer, you can construct a variety of different coordinate systems where any given non-inertial object is at rest. On the other hand, if you want the inertial frame where the center of your craft is at rest while the rest of it rotates around it, then your own rotation is irrelevant to calculating the velocities of the ships in this frame.

 

[Nishmaster]

Yes, the frame here I am referring to would be a rotating coordinate system, where every part of my craft is at rest.

I guess my more overarching question perhaps is this:

Ignore the fact that ships have mass for a second and assume Ship A and B are both traveling at 1c. My observer craft is at 1 light second from the collision point, and each ship is also 1 light second from the collision point. So now, I have an equilateral triangle between myself, the collision point, and Ship A, if that makes any sense.

I can find that at the start of this experiment I am 45 degrees from where I will be when the ships collide, thus my rotational velocity is roughly 14.32 rad/s. Since I am rotating with respect to Ship A, I would measure using Pythagoras that Ship A is 1.41 (sqrt(2)) light seconds away at the start, and 1 light second away at the end, and thus is moving toward me at .41 light seconds/s.

My question is: What would I measure as the speed of Ship B? Would it be greater than c?

Something I just realized is that I suppose my rotational velocity is not constant. It would have to be accelerating to keep Ship A stationary with respect to the x component of its vector.

 

[HallsofIvy]

Yes, the frame here I am referring to would be a rotating coordinate system, where every part of my craft is at rest.

Then, as Mintz114 said, you are not in an enertial frame of reference and the usual formulas for special relativity do not apply.

I guess my more overarching question perhaps is this:

Ignore the fact that ships have mass for a second and assume Ship A and B are both traveling at 1c. My observer craft is at 1 light second from the collision point, and each ship is also 1 light second from the collision point. So now, I have an equilateral triangle between myself, the collision point, and Ship A, if that makes any sense.

I can find that at the start of this experiment I am 45 degrees from where I will be when the ships collide

How do you do that? The angles in an equilateral triangle are 60 degrees, not 45 degrees.

, thus my rotational velocity is roughly 14.32 rad/s. Since I am rotating with respect to Ship A, I would measure using Pythagoras that Ship A is 1.41 (sqrt(2)) light seconds away at the start, and 1 light second away at the end, and thus is moving toward me at .41 light seconds/s.

My question is: What would I measure as the speed of Ship B? Would it be greater than c?

Something I just realized is that I suppose my rotational velocity is not constant. It would have to be accelerating to keep Ship A stationary with respect to the x component of its vector.

 

[Nishmaster]

Sorry, I have not yet had my coffee for the day. What I meant was a right triangle.