Special Relativity - Time Dilation:
A spaceship travels at 80% of the speed of light to a distant star 10 light-years away. Calculate the time elapsed for the spaceship's crew and the time elapsed on Earth during the journey.
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To calculate the time elapsed for the spaceship's crew and the time elapsed on Earth during the journey, we can use the time dilation equation from special relativity.
The time dilation equation is given by:
π‘β² = π‘β(1 β π£^2/π^2)
Where:
π‘β² is the time elapsed for the spaceship's crew
π‘ is the time elapsed on Earth
π£ is the velocity of the spaceship relative to the speed of light (π£ = 0.8π, where π is the speed of light)
Given that the distance to the star is 10 light-years, we can calculate the time elapsed on Earth using the formula:
π‘ = π/π£
= 10 light-years / (0.8π)
= 10 light-years / (0.8 * 299792458 m/s)
Using this speed for π£, we can convert the time elapsed on Earth to seconds:
π‘ = (10 * 365 * 24 * 60 * 60) s / (0.8 * 299792458 m/s)
Now, we can substitute this value of π‘ into the time dilation equation:
π‘β² = π‘β(1 β π£^2/π^2)
π‘β² = [(10 * 365 * 24 * 60 * 60) s / (0.8 * 299792458 m/s)] * β(1 β (0.8π)^2/π^2)
Evaluating this equation will provide the time elapsed for the spaceship's crew and the time elapsed on Earth during the journey.
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