Special Relativity and Kinematics Problem

[ToothandnaiL]

Homework Statement

There are three particles along the x-axis ordered from back to front, A, C, B. A travels at 4/5c and B travels at 3/5c, the speed of particle C is unknown. What must the speed of particle C be such that A and B will arrive at C at the same time? And, at what speed will both the particles approach C?

Homework Equations

(1) Δ(t)= ([Δ(t)]'/[1- (v^2/c^2)])^(1/2) (for time dilation in each reference frame)

>> $\gamma$= [1- (v^2/c^2)])^(1/2)

(2) u'=[(u-v)/(1- [uv/(c^2)]) (for speeds of particles relative to each other, u' is the modified speed of a relative moving object after special relativity is taken into account, u is the speed presented in the problem without reference to any other moving object or frame)

These are the two equations I have used to attempt this problem.

The Attempt at a Solution

I calculate the speed of particles A and B relative to each other using eq. 2. I split the problem into a reference frame for the A and B particles and a frame for the C particle. I use eq. 1 to account for time dilation and in turn speed differences among the particles. I think I have correctly calculated the speed of A and B to be approaching each other at 5/13 the speed of light. Other than that I have failed multiple times in trying to get the speed of C.
I've even gone as far as to assign a random length to the configuration of particles in order to have some concrete values to work with, although this shouldn't be necessary. If I know that both A and B are approaching C at 5/13c then I know there is a way to figure the speed of C. I think there is a kinematic aspect that I'm overlooking here. Any and all suggestions are welcome and thanked in advance.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Doc Al]

Are they all moving in the same direction? What are the initial locations of A, B, and C? Where do A and B meet? (All this can be done from the frame in which A, B, and C are moving at the given speeds.)

 

[ToothandnaiL]

Are they all moving in the same direction? What are the initial locations of A, B, and C? Where do A and B meet? (All this can be done from the frame in which A, B, and C are moving at the given speeds.)

All the particles are moving in the positive x direction. C is in between A and B, with behind and B in front. A and B will arrive at C's position simultaneously.

 

[Doc Al]

All the particles are moving in the positive x direction. C is in between A and B, with behind and B in front. A and B will arrive at C's position simultaneously.

How far apart are they when they start out?

 

[ToothandnaiL]

A and B are equidistant from C, sorry I forgot to mention this in the previous post.

 

[Doc Al]

A and B are equidistant from C, sorry I forgot to mention this in the previous post.

OK. To find the speed of C, here's what I would do. Just let L be the distance between A-C and C-B. Now just solve the kinematics problem of when will A overtake B? (Just solve it symbolically.) Then you can figure out the speed of C, since C has to meet B in that same time. This part has nothing really to do with relativity.

 

[ToothandnaiL]

Ok, that makes the problem a lot more straight forward. Thanks for bearing with me and your insightful input.