Calculating Time Dilation for Space Travelers

In summary, "Calculating Time Dilation for Space Travelers" explores the effects of relativistic speeds on time perception as described by Einstein's theory of relativity. It explains how time moves slower for individuals traveling at significant fractions of the speed of light compared to those remaining stationary on Earth. The article presents mathematical equations used to calculate the degree of time dilation based on velocity and gravitational influence, illustrating how this phenomenon can lead to astronauts aging more slowly than people on Earth during long space missions. The implications for future space travel and the potential for exploring distant galaxies are also discussed.
  • #1
stunner5000pt
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Homework Statement
A space traveler moving at a speed of 0.70c with respect to Earth makes a trip to a distant star that is stationary relative to Earth. He measures the length of this trip to be 6.5ly. What would be the length of this same trip (in ly) as measured by a traveler moving at a speed of 0.90c with respect to Earth?
Relevant Equations
[tex] \Delta t = \gamma t_{0} [/tex]
If we took the perspective of the space traveller themselves, they are stationary and the whole universe goes past them at 0.7c. THen th elapsed time of 6.5 yr looking outside is

[tex] \Delta t =6.5 \frac{1}{\sqrt{1-0.7^2}} = 9.11 yrs [/tex]

THen, when the traveller looks at the person travelling at 0.9c

the time taken should be

[tex] 9.11 \times \frac{1}{\sqrt{1-0.9^2}} = 3.97 yrs [/tex]

but the answer is 4.01 yr. Is this just down to rounding or is there something missing in the solution above?

thanks in advance for your help
 
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  • #2
stunner5000pt said:
... elapsed time of 6.5 yr ...
The problem does not say anything about such time, 6.5 yr. It describes a distance, 6.5 ly.
 
  • #3
Hill said:
The problem does not say anything about such time, 6.5 yr. It describes a distance, 6.5 ly.
Aha! missed that, thank you

ok so then that means the distance travelled according to the traveller is:

[tex] 6.5c \times 24 \times 365 \times 60^2 = 6.14 \times 10^{16} m [/tex]

when the person is in motion, how is the 6.5 ly being interpreted ? Is the noninertial observer seeing a shorter'length that the stationary length i.e. is the above number the length observed from rest?
 
  • #4
No need to convert or to interpret ly. It is a unit of distance as good as any.
I don't see a "noninertial observer" in the problem. However, there is length contraction rather than time dilation here.
 
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  • #5
stunner5000pt said:
ok so then that means the distance travelled
The distance travelled by the Earth and the destination planet.

And as has already been said, ly is a perfectly fine length unit. Preferable to meters here really as by definition c = 1 ly/y.
 

FAQ: Calculating Time Dilation for Space Travelers

What is time dilation?

Time dilation is a concept in Einstein's theory of relativity that describes how time passes at different rates for observers in different frames of reference. For space travelers moving at high velocities relative to an observer on Earth, time appears to pass more slowly.

How do you calculate time dilation for space travelers?

Time dilation can be calculated using the Lorentz factor, which is defined as γ = 1 / √(1 - v²/c²), where v is the velocity of the space traveler relative to the speed of light c. The time experienced by the space traveler (t') is related to the time experienced by an observer on Earth (t) by the equation t' = t / γ.

What is the significance of the speed of light in time dilation calculations?

The speed of light (c) is a fundamental constant in the equation for time dilation. It serves as the maximum possible speed in the universe and is the reference point for calculating how much time slows down for a moving object. As an object's velocity (v) approaches the speed of light, the time dilation effect becomes more pronounced.

At what speeds does time dilation become noticeable?

Time dilation effects become noticeable at speeds that are a significant fraction of the speed of light, typically greater than 10% of c. At everyday speeds, such as those experienced in cars or airplanes, the effects are minuscule and virtually undetectable.

How does time dilation affect long-duration space missions?

For long-duration space missions traveling at high velocities, time dilation means that astronauts would age more slowly compared to people on Earth. This has implications for mission planning, communication, and the health of astronauts, as their biological clocks would be running at a different rate compared to those of their Earth-bound counterparts.

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