# Should I use time dilation or length contraction?

• Joao Victor Dantas
In summary: According to rocket 1, how long before they collide? According to rocket 1, it will take it 88.9 minutes to reach the point of collision.
Joao Victor Dantas
1. Homework Statement [/B]

This is a problem that was in my Physics HW.

Two powerless rockets are on a collision course. The rockets are moving with speeds of 0.800c and 0.600c and are initially ## 2.52 × 10^{12} ## m apart as measured by Liz, an Earth
observer, as shown in Figure P1.59. Both rockets are 50.0 m in length as measured by
Liz. (a) What are their respective proper lengths? (b) What is the length of each rocket as
measured by an observer in the other rocket? (c) According to Liz, how long before the
rockets collide? (d) According to rocket 1, how long before they collide? (e) According
to rocket 2, how long before they collide? (f) If both rocket crews are capable of total
evacuation within 90 minutes (their own time), will there be any casualties?

My doubt is on letters (d) and (e). I don't know if I am supposed to apply the time Lorentz transformation using the value obtained in (c) or if I should calculate this time based on the speed each rocket sees the other approaching and the distance using length contraction. I found two answers on the internet.

## Homework Equations

## L = L_{0}\sqrt {1 - \frac {v^2} {c^2}} ##
## \Delta t' = \frac 1 {\sqrt {1 - \frac {v^2} {c^2}}} \Delta t ##
## V' = \frac {u - V_x} {1 - \frac {uV_x} {c^2}} ##

## The Attempt at a Solution

By using the mentioned equations, I obtained that (a) ## L_1 = 83.3 m ## and ## L_2 = 62.5 m ## . (b) ## L_1 = 27.0 m ## in the frame of rocket 2 and ## L_2 = 21.0 m ## in the frame of rocket 1. (c) ## \frac {\Delta S} {v_1 + v_2} = 6000 sec = 100 min ## .

When it comes to letter (d) that something goes wrong. my first approach to it was to use the length contraction observed by 1 and divide it by the speed 1 sees 2 approaching. ## L = L_{0}\sqrt {1 - \frac {v^2} {c^2}} = 2.52 \times 10^{12} \times 0.6 = 1.512 \times 10^{12} ## and ## V' = \frac {u - V_x} {1 - \frac {uV_x} {c^2}} = \frac { 0.8c - ( - 0.6c)} {1 - \frac { (- 0.48c^2)} { c^2 }} = 0.945c ## . Dividing these results we have ## \frac {L} {V'} = 5,333 sec = 88.9 min ## . Although, using ## \Delta t' = \frac 1 {\sqrt {1 - \frac {v^2} {c^2}}} \Delta t ## , where t' is Liz's time of 100 min, we obtain ## 100 min = 1.6666 \times \Delta t ## and ## \Delta t = 60 min ##. This same problem happens when I try to solve (e), and I've taken a look at several solutions on the internet, being half of them solved in the first way, and half in the second. Shouldn't these results agree? If not, why?

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As with most relativity problems, the difficulty is with the relativity of simultaneity. According to Liz on Earth, the start event for both Liz and L1 is simultaneous. The end event is a collision and is naturally simultaneous for all parties involved. According to L1, the start event for L1 is not simultaneous with the start event for Liz.

Joao Victor Dantas
This makes sense. So 88.9 min would be the time it takes for the observer in rocket 1 to get to the point of collision from the start of HIS measurement and 60 min would be the time is takes for this observer from the start of Liz's measurement, correct?

Can you draw a position-vs-time graph of the problem?

Joao Victor Dantas said:
This makes sense. So 88.9 min would be the time it takes for the observer in rocket 1 to get to the point of collision from the start of HIS measurement and 60 min would be the time is takes for this observer from the start of Liz's measurement, correct?
I am not sure that I understand your phrasing here. Consider that L1 has a stopwatch. He starts it at some point. And we are asked for its reading at the event of the collision.

At what event does L1 start his stopwatch? I would suggest having him start it at the event that Liz considers to be simultaneous with the scenario start.

Maybe it would help to resolve the issue if you calculated the "contracted" distance that
each ship travels and then the time to traverse this distance.
Use the 100 min. that you calculated as measured by Liz.

## 1. What is time dilation and length contraction?

Time dilation and length contraction are two concepts in the theory of relativity that describe how time and space are affected by an object's speed and acceleration. Time dilation refers to the slowing down of time as an object moves closer to the speed of light, while length contraction refers to the shortening of an object's length in the direction of its motion.

## 2. When should I use time dilation or length contraction?

The use of time dilation or length contraction depends on the specific situation and what you are trying to measure. In general, time dilation is more relevant for objects moving at high speeds, while length contraction is more relevant for objects accelerating at high rates. It is important to consider both concepts when studying fast-moving objects.

## 3. What are the practical applications of time dilation and length contraction?

Time dilation and length contraction have significant practical applications in fields such as astrophysics, aerospace engineering, and particle physics. They are essential for understanding the behavior of high-speed objects and have been confirmed through numerous experiments and observations.

## 4. How do time dilation and length contraction affect our everyday lives?

While time dilation and length contraction may seem like abstract concepts, they do have real-world implications. For example, GPS satellites must account for the effects of time dilation in order to accurately calculate positions on Earth. Additionally, particle accelerators, such as the Large Hadron Collider, rely on length contraction to achieve high speeds.

## 5. Are time dilation and length contraction proven theories?

Time dilation and length contraction are well-established theories that have been extensively tested and confirmed through experiments and observations. They are integral components of the theory of relativity, which has been one of the most successful and influential theories in modern physics. However, like all scientific theories, they are subject to further research and refinement as our understanding of the universe continues to evolve.

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