Answer:
The time dilation effect can be calculated using the time dilation formula, which is derived from the theory of special relativity. The formula is: Δt' = Δt / √(1 - v²/c²), where Δt is the time interval measured by a stationary observer, Δt' is the time interval measured by an observer in motion, v is the relative velocity between the two observers, and c is the speed of light.
In this case, Valery Polyakov spent 438 consecutive days orbiting the Earth on the Mir Space station at an orbital velocity where the gamma factor is 1.000000000338. The gamma factor is equal to 1 / √(1 - v²/c²), so we can solve for v²/c² = 1 - (1 / gamma)² = 2.28 x 10⁻¹⁸. Plugging this into the time dilation formula, we get that Δt' = Δt / √(1 - 2.28 x 10⁻¹⁸) ≈ Δt (1 + 1.14 x 10⁻¹⁸).
The time interval measured by a stationary observer on Earth is 438 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 37,843,200 seconds. Plugging this into the formula above, we get that Δt' ≈ 37,843,200 seconds (1 + 1.14 x 10⁻¹⁸) ≈ 37,843,200.000043 seconds.
Therefore, the difference in elapsed time between the stopwatch on Earth and the stopwatch on Mir over the duration of Valery Polyakov's flight would be approximately 0.000043 seconds or **4.3 x 10⁻⁵ seconds**. This answer is not among the answer choices you provided.