Relativity, time dilation

In summary, in 1962, a person orbited the Earth 22 times and the press reported that each orbit caused the person to age 2 millionths of a second less than if they remained on Earth. Assuming the person was 160km above the Earth in a circular orbit, the time difference between the person and someone on Earth for 22 orbits is approximately 2 microseconds, as stated in the press report. This takes into account the effects of special relativistic and gravitational time dilation.
  • #1
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Homework Statement


in 1962 a person orbited the Earth 22 times, the press stated that each orbit he aged 2millionths of a second less, then if he remained on earth.

a.assuming he was 160km above the earth, in a circular orbit, determine the time difference between on Earth and the orbiting astronaut for 22orbits.


Homework Equations


V=root(GM/R)
tau=gamma*t

The Attempt at a Solution


v=root(6.67x10^-11*5.98x10^24/6.37x10^6+160x10^3)=7815.5m/s

distance(circumference)=2*pi*r=2*pi*(6.37x10^6+160x10^3)=4.1x10^7m

t=4.1x10^7/7815.5=5249.7s

gamma =approx (1+.5*(7815.5/3x10^8))=1.000000001

tau=gamma*t=5249.719894s

tau-t=3.563x10^-6s? is this rite, because, i can't see anywhere i have gone wrong... used exact answers in all my calculations. i guess that is kind of close to 2x10^-6s
 
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  • #2
fredrick08 said:

Homework Statement


in 1962 a person orbited the Earth 22 times, the press stated that each orbit he aged 2millionths of a second less, then if he remained on earth.

a.assuming he was 160km above the earth, in a circular orbit, determine the time difference between on Earth and the orbiting astronaut for 22orbits.


Homework Equations


V=root(GM/R)
tau=gamma*t

The Attempt at a Solution


v=root(6.67x10^-11*5.98x10^24/6.37x10^6+160x10^3)=7815.5m/s

distance(circumference)=2*pi*r=2*pi*(6.37x10^6+160x10^3)=4.1x10^7m

t=4.1x10^7/7815.5=5249.7s

gamma =approx (1+.5*(7815.5/3x10^8))=1.000000001

tau=gamma*t=5249.719894s

tau-t=3.563x10^-6s? is this rite, because, i can't see anywhere i have gone wrong... used exact answers in all my calculations. i guess that is kind of close to 2x10^-6s
What was the relative speed of the astronaut relative to Houston? I think you have to factor in the rotation of the Earth here.

AM
 
  • #3
fredrick08 said:

Homework Statement


in 1962 a person orbited the Earth 22 times, the press stated that each orbit he aged 2millionths of a second less, then if he remained on earth.

a.assuming he was 160km above the earth, in a circular orbit, determine the time difference between on Earth and the orbiting astronaut for 22orbits.


Homework Equations


V=root(GM/R)
tau=gamma*t

The Attempt at a Solution


v=root(6.67x10^-11*5.98x10^24/6.37x10^6+160x10^3)=7815.5m/s

distance(circumference)=2*pi*r=2*pi*(6.37x10^6+160x10^3)=4.1x10^7m

t=4.1x10^7/7815.5=5249.7s

gamma =approx (1+.5*(7815.5/3x10^8))=1.000000001

tau=gamma*t=5249.719894s

tau-t=3.563x10^-6s? is this rite, because, i can't see anywhere i have gone wrong... used exact answers in all my calculations. i guess that is kind of close to 2x10^-6s

I think you might have to fix up your gamma, the calculation for gamma should be
gamma = 1 / SQRT(1 - v^2/c^2)) which is going to be slightly different from your answer.

The formula for t = t0*gamma and you should get hopefully the dilated time if everything
is relative.
 
  • #5
fredrick08 said:
tau-t=3.563x10^-6s? is this rite, because, i can't see anywhere i have gone wrong... used exact answers in all my calculations. i guess that is kind of close to 2x10^-6s
You are looking at only part of the time dilation. The astronaut ages less because of special relativistic and gravitational time dilation.

Fortunately, you don't need to know either special or general relativity to answer this question because you have already been given the answer to this question. All you need to compute is how much time passed for a person on the surface of the Earth and use the given relativistic time dilation to compute how much time passed on-orbit.
 
  • #6
We can see that this is non-relativisitic speed so we can approximate gamma by 1 + v^2/2c^2. I think that is where you went wrong. You were approximating by 1+v/2c.

The orbital speed is 7816 m/sec as you found. So the orbital period is 5.25 x10^3 seconds as you have found. 22 periods amounts to 1.14 x 10^5 seconds or about 32 hours. I don't see the 22 orbits in your calculation.

But the Earth is also rotating, and during that period a person on the Earth is moving at a speed of 460 m/sec. So the relative speed (depending on the direction of the orbit) is a range from (7816 + 460) m/sec to (7816 - 460) m/sec. (8276 m/sec to 7356 m/sec)

From this I get the maximum time dilation to be:

[tex]t'-t = 4.2 x 10^{-5} [/tex] sec

The minimum time dilation is about 3.4 x 10^-5 seconds.

AM
 
  • #7
You are forgetting gravitational time dilation. A person 160 km above the surface of the Earth experiences less gravitational time dilation than someone on the surface of the Earth.

There is no reason to calculate the time dilation at all because the problem already gives that answer: 2 microseconds per orbit.
 
  • #8
D H said:
You are forgetting gravitational time dilation. A person 160 km above the surface of the Earth experiences less gravitational time dilation than someone on the surface of the Earth.

There is no reason to calculate the time dilation at all because the problem already gives that answer: 2 microseconds per orbit.
The first rule of being a scientist should be: don't to believe what you read in the press about science.

I think the question is asking the student to determine whether the press report was right. It appears to be right.

AM
 

1. What is relativity?

Relativity is a theory proposed by Albert Einstein in the early 20th century. It states that the laws of physics are the same for all observers, regardless of their relative motion. This means that there is no absolute frame of reference and that the perception of space and time can vary depending on the observer's perspective.

2. What is time dilation?

Time dilation is a phenomenon predicted by the theory of relativity. It states that time passes slower for objects that are moving at high speeds or experiencing strong gravitational forces. This means that a person on a fast-moving spaceship will experience time passing slower compared to someone on Earth.

3. How does time dilation affect everyday life?

While time dilation is a real phenomenon, its effects are only noticeable at extremely high speeds or in strong gravitational fields. For example, the time dilation experienced by astronauts in space is only a few milliseconds per day. It has no significant impact on our daily lives.

4. Is time travel possible with time dilation?

Time travel is a popular concept in science fiction, but it is not possible with time dilation. While an object may experience time passing slower, it cannot travel back in time or skip ahead. Time dilation is only a change in perception of time, not actual time travel.

5. What evidence supports the theory of relativity?

There is a wealth of evidence that supports the theory of relativity, including the observed bending of starlight by the Sun's gravitational field, the precision of GPS satellite navigation, and the results of particle collider experiments. The theory has been extensively tested and has consistently been shown to accurately describe the behavior of the universe.

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