A spaceship from another galaxy passes over the solar system directly above a radial line from the sun to the Earth. (We measure that distance to be 1.5 × 10¹¹ m.) On Earth, the spaceship is observed to be traveling at a speed of 0.80 c, for which γ = 5/3. As measured on Earth it takes the spaceship 625 seconds to travel from the sun to Earth. When a scientist in the spaceship measures the Earth-sun distance and the time it takes her to travel that distance, she finds the results are respectively A) 9.0 × 10¹⁰ m; 375 s. B) 9.0 × 10¹⁰ m; 625 s. C) 1.5 × 10¹¹ m; 625 s. D) 2.5 × 10¹¹ m; 625 s. E) 2.5 × 10¹¹ m; 1042 s.

Question

Quizplus · Accepted Answer

The correct answer is based on the concepts of time dilation and length contraction in special relativity. The length as measured in the spaceship's frame will be contracted compared to the length measured on Earth due to its high velocity (length contraction). The formula for length contraction is $L = L_0/\gamma$, where $L_0$ is the rest length (1.5 × 10^11 m), and $\gamma = 5/3$. This gives $L = (1.5 × 10^11 m) / (5/3) = 9.0 × 10^10 m$, which matches option A for the length. The time dilation formula is $t = \gamma t_0$, where $t_0$ is the proper time. Since the spaceship is moving at a speed where $\gamma = 5/3$, and it takes 625 seconds as measured on Earth, the time measured in the spaceship would actually be shorter because it's the proper time interval. However, the calculation provided directly for the time in the question seems to contradict the expected outcome of time dilation, suggesting a misunderstanding in the application of the concept. The correct interpretation of the given information leads to the length being contracted as described, but the time aspect in the options provided does not align with the expected outcome of time dilation, which would typically result in a longer time measured from the Earth frame compared to the spaceship. The correct answer, based on length contraction alone, is A, but the explanation for time requires careful consideration of the relativistic effects, which seem to be inconsistently applied in the provided options.

A Spaceship from Another Galaxy Passes Over the Solar System