Calculate the time dilation experienced by astronauts traveling at 99.999% of the speed of light on a journey to a distant star 10 light-years away, as measured by Earth observers.
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To calculate the time dilation experienced by the astronauts traveling at 99.999% of the speed of light, let's use the time dilation formula:
Δt' = Δt / (√(1 - (v^2/c^2)))
Where:
Δt' = time experienced by the astronauts
Δt = time measured by Earth observers
v = velocity of the astronauts (99.999% the speed of light)
c = speed of light (299,792,458 meters per second)
We can convert the distance to meters:
Distance = 10 light-years = 10 * (9.461 × 10^15 meters) = 9.461 × 10^16 meters
Now, we can plug these values into the time dilation formula:
Δt' = Δt / (√(1 - (v^2/c^2)))
Δt' = 10 light-years / (√(1 - ((0.99999c)^2/c^2)))
Δt' = 9.461 × 10^16 meters / (√(1 - (0.99999^2)))
Δt' = 9.461 × 10^16 meters / (√(1 - 0.9999999998))
Δt' = 9.461 × 10^16 meters / (√0.0000000001999)
Δt' = 9.461 × 10^16 meters / 0.00044721
Δt' = 2.114 × 10^20 seconds
Therefore, the time dilation experienced by the astronauts would be approximately 2.114 × 10^20 seconds.
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