How Do Time Dilation and Length Contraction Affect Colliding Rockets?

In summary, the conversation discusses the problem of two powerless rockets on a collision course and addresses the questions of their respective proper lengths, their lengths as measured by an observer on the other rocket, and the time of collision according to different observers. The equations of Lorentz length and time transformations are used to solve the problem.
  • #1
Tyst
26
0

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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  • #2
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...

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

Also, remember the form of equation (2) changes for rocket 2.
 
  • #3
Thanks very much for your help neutrino - got it sorted now!
 

FAQ: How Do Time Dilation and Length Contraction Affect Colliding Rockets?

1. What is time dilation?

Time dilation is a phenomenon in which time is experienced at different rates for observers in different frames of reference. This means that time can appear to move slower or faster for individuals depending on their relative motion and gravitational fields.

2. How does time dilation occur?

Time dilation occurs due to the principles of special and general relativity. According to these theories, time and space are not absolute but are instead relative to the observer's frame of reference. This means that the passage of time and the measurement of distances can be affected by factors such as speed and gravity.

3. What is length contraction?

Length contraction is a concept related to time dilation, but instead of time, it applies to the physical length of an object. According to the theory of relativity, an object's length will appear shorter to an observer who is in motion relative to the object. This effect is only noticeable at high speeds near the speed of light.

4. How does length contraction work?

Length contraction occurs because space and time are not independent but are instead interconnected. As an object moves closer to the speed of light, its length in the direction of motion will appear to shorten from the perspective of an outside observer. This is due to the fact that time is also perceived to move slower for the moving object, causing the length to appear to contract.

5. Can time dilation and length contraction be observed in everyday life?

Yes, time dilation and length contraction can be observed in everyday life, but the effects are only noticeable at extremely high speeds or in the presence of strong gravitational fields. For example, the Global Positioning System (GPS) must take into account time dilation in order to function accurately. Additionally, particles in particle accelerators experience length contraction as they approach the speed of light. However, these effects are not noticeable in our daily lives at typical speeds and gravitational fields.

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