First Year Relativity Problem from Exam

In summary, the TV signals from Earth and the Planet of the Apes were sent to each other at different times but were considered to be simultaneous by the observers on Earth. The spacetime interval between the two events was negative, and the speed of the observer flying on the rocket was 0.70c.
  • #1
Heresy
54
0

Homework Statement


The Earth and the Planet of the Apes are in a reference frame where they are stationary relative to one another, and are measured to be 2 light years apart. The observers on Earth send a TV signal to the Planet of the Apes with a picture of a banana. 0.6 years later, the observers on Earth receive a TV signal from the Planet of the Apes saying "Give us the banana!".

Question 1: What is the minimum speed that the TV signals must travel to fulfill the above conditions? (This was not the original wording but feel free to ignore this question, I don't quite remember it anyway)
Question 2: What is the spacetime interval between the two events, as measured by an observer on Earth?
Question 3: What is the spacetime interval between the two events as measured by an observer on a rocket traveling from Earth to PoA who finds that the two events (Earth and PoA sending their respective TV signals) are simultaneous?
Question 4: According to the observer, what is the distance between the two planets?
Question 5: At what speed is the observer moving relative to Earth and the PoA?

The Attempt at a Solution



Event 1: Earth sends TV signal to PoA
Event 2: PoA sends TV signal to Earth

Most of the people in my class (including me) assumed that one event caused the other, but is that possible if an observer in a reference frame measures them to be simultaneous?

If I take Event 2 to happen at t = -1.4 years and Event 1 to happen at t = 0 years (for the signal from PoA to reach the Earth at t = 0.6 years), the spacetime interval turns out to be a negative number. Answer to Question 4 turns out to be 1.4 x 10^16 meters, and the answer to Question 5 turns out to be 0.63c

Was I on the right track?
 
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  • #2
Heresy said:
Most of the people in my class (including me) assumed that one event caused the other, but is that possible if an observer in a reference frame measures them to be simultaneous?
How can these events be causally related? The signal from 1 won't even reach PoA until long after the the signal from 2 has arrived at earth. (But you're correct that if are simultaneous in some frame, then they can't be causally related.)

If I take Event 2 to happen at t = -1.4 years and Event 1 to happen at t = 0 years (for the signal from PoA to reach the Earth at t = 0.6 years), the spacetime interval turns out to be a negative number. Answer to Question 4 turns out to be 1.4 x 10^16 meters, and the answer to Question 5 turns out to be 0.63c
Show how you arrived at these results.

(Planet of the Apes is the earth! :eek:)
 
  • #3
Most people assumed a causal relation because Question 1 implied that a signal went to the PoA and back within a time interval of 0.6 years... as I said earlier I don't remember the exact wording of this question so I'll just let it go.

Operating under the Earth's reference frame here
-Distance between Earth and PoA is 2 light years, so for the signal from PoA asking for the banana to reach the Earth at t = 0.6 years, PoA must have emitted their TV signal at t = -1.4 years. Earth emitted their signal at t = 0 years

-Spacetime interval squared = c^2 (deltaT)^2 - (deltaX)^2
-Converted light years to metres and years to seconds
-Setting deltaT (1.4 years or 44150400 seconds) and deltaX (1.89 x 10^16 m) yields a result of s^2 = -1.83 x 10^32 metres squared. (Answer for question 2 and 3)

For the observer to measure them as simultaneous, I used the invariant spacetime interveral by setting deltaT = 0 and solving for deltaX, which came out to be 1.35 x 10^16 metres (Answer for question 4).

To find the speed of the observer flying on the rocket, I used the formula for length contraction

1.35 x 10^16 = sqrt(1-v^2/c^2)1.89 x 10^16

With a little bit of algebra, v turns out to be 0.70c (answer for question 5)

Um... I guess the minor difference in numbers came about when I was fiddling with the number of digits I was handling with - but is my method correct?

NB: I didn't write this for my exam on Wednesday morning, but it doesn't even matter to me anymore - I just want to know how to do it properly.
 
  • #4
Heresy said:
Um... I guess the minor difference in numbers came about when I was fiddling with the number of digits I was handling with - but is my method correct?
Perfectly correct.
 
  • #5
Alright, thanks a lot!
 

What is the "First Year Relativity Problem from Exam"?

The "First Year Relativity Problem from Exam" is a commonly used exam question in introductory physics courses that tests students' understanding of Einstein's theory of relativity. It typically involves a scenario where an observer measures the speed of light from two different frames of reference and is asked to calculate the relative velocity between the frames.

Why is the "First Year Relativity Problem from Exam" important?

The "First Year Relativity Problem from Exam" is important because it helps students understand the fundamental principles of Einstein's theory of relativity, which has had a profound impact on our understanding of the universe. It also tests students' ability to apply mathematical concepts to real-world scenarios, which is a crucial skill in the field of science.

What is the formula used to solve the "First Year Relativity Problem from Exam"?

The formula used to solve the "First Year Relativity Problem from Exam" is the relativistic velocity addition formula, which states that the relative velocity between two frames of reference is equal to the difference of their velocities divided by 1 plus the product of their velocities and the speed of light squared.

What are some common mistakes students make when solving the "First Year Relativity Problem from Exam"?

Some common mistakes students make when solving the "First Year Relativity Problem from Exam" include forgetting to convert units, using the wrong formula, or not properly applying the concept of relative velocity. It is important for students to carefully read the problem and double-check their calculations to avoid these errors.

How can students prepare for the "First Year Relativity Problem from Exam"?

To prepare for the "First Year Relativity Problem from Exam," students should review the basic principles of Einstein's theory of relativity and practice solving similar problems. They should also make sure they have a solid understanding of the relativistic velocity addition formula and how to apply it in different scenarios. Seeking help from a tutor or attending review sessions can also be beneficial in preparing for this type of problem.

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