How Does the Twin Paradox Illustrate Time Dilation and Length Contraction?

[Aquafina20]

[SOLVED] Intro course Twin Paradox

Homework Statement

You probably already know. One twin stays on earth. One goes 4ly away at .8c. The question asked a bunch of different things. How long does the twin on the ship think it takes. How long does the twin on Earth think it takes. How long does the twin on the ship measure the trip to be (length not time).

Homework Equations

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The Attempt at a Solution

My issue is conceptual. The proper time is the frame where both events occur at the same place correct? So shouldn't the twin on the ship measure proper time of 4ly/.8c = approx. 5 years. So then time dilation says on Earth they'd measure 5years/.6 or about 8.33 years? But then they say what length does the twin on the ship measure. Wouldn't it be 4ly? Because for 5 years he travels at .8c? But Earth would also measure 4 ly would it not?

 

[anathema]

Thank you for bringing up this interesting topic! The Twin Paradox is a classic thought experiment in special relativity that can help us understand the concept of time dilation. Let's break down the problem and address each question separately.

First, let's define the scenario. We have two twins, A and B. Twin A stays on Earth, while Twin B travels away from Earth at a speed of 0.8c (80% of the speed of light) for a distance of 4 light years. We will also assume that Twin B turns around and returns to Earth at the same speed.

Now, let's address the first question: How long does the twin on the ship think it takes? This question is asking for the time measured by Twin B, who is on the ship traveling at 0.8c. In this frame of reference, the distance traveled is 4 light years and the speed is 0.8c. Using the equation t = d/v, we can calculate the time to be 5 years. This is the proper time, as it is measured in the frame where the events (departure and return) occur at the same place.

Next, we can answer the second question: How long does the twin on Earth think it takes? This question is asking for the time measured by Twin A, who is on Earth. In this frame of reference, the distance traveled is also 4 light years, but the speed is now 0.6c (since Twin B is traveling at 0.8c and special relativity tells us that velocities do not add up in a simple way). Using the same equation, we can calculate the time to be approximately 8.33 years. This is longer than the time measured by Twin B because of time dilation, which states that time appears to pass slower for objects moving at high speeds.

Finally, we can address the third question: How long does the twin on the ship measure the trip to be (length not time)? This question is asking for the distance measured by Twin B. In this frame of reference, the time is still 5 years, but the speed is now 0.8c. Using the same equation, we can calculate the distance to be 4 light years. This is the same distance measured by Twin A, as distances do not change with relative motion.

I hope this helps clarify the concepts involved in the Twin Paradox. Keep up the good

 

[ogb p]

The Twin Paradox is a thought experiment that explores the concept of time dilation in special relativity. It is often used to demonstrate the effects of time dilation on objects traveling at high speeds. In this scenario, one twin stays on Earth while the other twin travels away from Earth at a high speed and then returns. The paradox arises when the two twins' experiences of time and space are compared.

To answer the questions posed, we first need to understand the concept of proper time. Proper time is the time measured by an observer who is at rest with respect to the events being measured. In this case, the twin on the ship is the observer who is moving at a constant velocity and is therefore at rest with respect to the events of the trip. Therefore, the proper time for the twin on the ship is 5 years, as you correctly calculated.

The twin on Earth, however, is not at rest with respect to the events of the trip. They are observing the twin on the ship moving at a high speed. According to the time dilation equation, the twin on Earth will measure a longer time for the trip, approximately 8.33 years, as you mentioned.

As for the length measured by the twin on the ship, it is important to note that length contraction is also a factor in special relativity. This means that objects in motion appear shorter in the direction of motion to an observer at rest. Therefore, the twin on the ship will measure the distance traveled to be shorter than 4 light years. This can be calculated using the length contraction equation, which takes into account the relative velocity between the two observers.

In conclusion, the twin paradox highlights the effects of time dilation and length contraction in special relativity. It is important to keep in mind that these effects are relative to the observer's frame of reference and can lead to seemingly contradictory results. However, they have been extensively tested and confirmed by experiments, and are essential for our understanding of the universe.