How Do Time Dilation and Length Contraction Affect Colliding Rockets?

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SUMMARY

The discussion focuses on the relativistic effects of time dilation and length contraction on two rockets, moving at speeds of 0.800c and 0.600c, respectively. The proper lengths of the rockets are calculated as 83.33m for ship 1 and 62.50m for ship 2. Lengths observed from each rocket's frame are determined using the Lorentz velocity transformation, yielding 27.03m for ship 1 as observed by ship 2 and 20.27m for ship 2 as observed by ship 1. The correct application of the Lorentz time transformation for calculating collision time is emphasized, with the corrected equation being t' = γ(t - (xv/c²)).

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  • Understanding of special relativity concepts, including time dilation and length contraction.
  • Familiarity with Lorentz transformations and their equations.
  • Knowledge of relativistic velocity transformations.
  • Ability to perform calculations involving speeds as fractions of the speed of light (c).
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Homework Statement



Two powerless rockets are on a collision course. They are moving with speeds 0.800c (ship 1) and 0.600c (ship 2) and are initially 2.25*10^12 m apart. The ships are both 50m in length as observed by a stationary observer on earth.

a) what are their respective proper lengths?
b) what is the length of each rocket as measured by an observer on the other rocket?
c) How long before the rockets collide according the observer on earth, rocket 1 and rocket 2?

Homework Equations



(1) L=Lp(1-v^2/c^2)^(1/2) (Lp = proper length)
(2) t'=(1/(1-v^2/c^2)) (t-v/c^2x) (Lorentz time transformation
(3) u'=(u-v)/(1-(uv/c^2)) (Relativistic velocity transformation)

(apologies for the clumsy notoation)

The Attempt at a Solution


I've solved a and found the proper lengths to be 83.33m for ship 1 and 62.50 for ship 2.

I have also solved b, though I'm not 100% sure of my answer:

Using the Lorentz velocity transform (equation 3 above)
(-.6c-.8c)/[1-(-.6c*.8c)/c^2] = -2.838*10^8 m/s
I've taken ship 1 as reference frame S' and ship 2 is traveling in the -ve x direction according to 1, hence the -.6c and the -ve answer.
Substituting -2.838*10^8 into the Length contraction formula (eq 1) the length of ship 1 obswerved by ship 2 was 27.03m and ship2 observed by ship1 was 20.27m.
Is this correct?

As for part c, i am not sure how to apply the Lorentz transform in each case... In fact, I'm very confused! Is equation 2 (above) the correct one to apply?

---
Also, is a question like this considered advanced undergrad? Or would i be better posting it in the other thread?

Thank you in advance for your help
 
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Your answers for a and b seem to be correct. As for c, calculate the time of collision and the point in space where it occurs from the Earth frame (choose suitable origin). Use that as your t and x in equation (2)(which is actually wrong).

The x should be in the numerator...

t' = \gamma\left(t - \frac{xv}{c^2}\right)

Also, remember the form of equation (2) changes for rocket 2.
 
Thanks very much for your help neutrino - got it sorted now!
 

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