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Rod R1 has a rest length `1m and rod R2 has rest length 2m. R1 is moving towards right with velocity v and R2 is moving towards left with velocity v with respect to lab frame. If R2 has length 1m in rest frame of R1, then v/c is what?

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I let vel of R1 be u, of R2 be w w.r.t lab and velocity of R2 w.r.t. R1 be w'.

Then u = v, and w = -v.

Then by relativistic velocity addition formula:

w' = (w - u)/(1- (u*w)/c^2)

=> w' = -2v/(1+v^2/c^2) -----------(1)

Now, according to Lorentz contraction:

L/L0 = √(1 - (w'/c)^2 )

But L/L0 = 1/2

Solving, I find w'/c = (√3)/2 ------------(2)

From simplifying (1) and (2):

v^2 + (4/√3) vc + c^2 = 0

So, v = -c/√3 or v = -√3 *c.

But... why is answer negative?

Even if I reject the 2nd one and accept the magnitude of first, I get v = 0.577c...

But the exact answer given is v = 0.6c.

Where did I go wrong?

Also, the solution given says:

Where did this one come from??

Any help will be appreciated.

-----------------------

I let vel of R1 be u, of R2 be w w.r.t lab and velocity of R2 w.r.t. R1 be w'.

Then u = v, and w = -v.

Then by relativistic velocity addition formula:

w' = (w - u)/(1- (u*w)/c^2)

=> w' = -2v/(1+v^2/c^2) -----------(1)

Now, according to Lorentz contraction:

L/L0 = √(1 - (w'/c)^2 )

But L/L0 = 1/2

Solving, I find w'/c = (√3)/2 ------------(2)

From simplifying (1) and (2):

v^2 + (4/√3) vc + c^2 = 0

So, v = -c/√3 or v = -√3 *c.

But... why is answer negative?

Even if I reject the 2nd one and accept the magnitude of first, I get v = 0.577c...

But the exact answer given is v = 0.6c.

Where did I go wrong?

Also, the solution given says:

Where did this one come from??

Any help will be appreciated.

**[Moderator's note: Moved from a technical forum and thus no template.]**
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