According to the principles of Einstein's theory of relativity, as an object with mass approaches the speed of light, its relativistic mass increases, and it requires an infinite amount of energy to accelerate it to the speed of light. Therefore, an object with mass, like a spaceship with humans on board, cannot reach or exceed the speed of light.
However, we can still calculate the time it would take for an observer on Earth to see a spaceship traveling at a significant fraction of the speed of light covering a distance of one light-year.
Let's consider a spaceship traveling at 99% of the speed of light (0.99c). Using the Lorentz factor, which is a term used in special relativity to calculate time dilation and length contraction, we can find the time experienced by the observer on Earth:
Time dilation factor (γ) = 1 / sqrt(1 - v^2/c^2)
Where: v = velocity of the spaceship (0.99c) c = speed of light in a vacuum
γ = 1 / sqrt(1 - 0.99^2) ≈ 7.0888
Now, let's calculate the time it would take for the spaceship to cover one light-year (approximately 9.461 trillion kilometers) from the perspective of the observer on Earth:
Time taken = (Distance) / (Velocity) Time taken ≈ 9.461 trillion km / (0.99c) ≈ 9.535 years
So, from the perspective of the observer on Earth, it would take approximately 9.535 years for the spaceship traveling at 99% of the speed of light to cover a distance of one light-year. However, for the travelers on board the spaceship, less time would have passed due to time dilation, following the Lorentz factor calculated earlier.