Question

# Sophia (on Earth) sees her twin sister, Stella, passing by on her spacecraft at 0.70cc and...

Sophia (on Earth) sees her twin sister, Stella, passing by on her spacecraft at 0.70cc and observes that Stella’s clock runs slower than her own. Sophia’s favorite movie runs for 1.90 hours. How long by Sophia’s clock, in hours, would the show run for Stella? (Include at least two decimal places in your answer.)

To calculate the time dilation effect due to the relative motion between Sophia and Stella, we can use the time dilation formula:

t' = t / sqrt(1 - v^2/c^2)

Where: t' is the time experienced by Stella (in her reference frame) t is the time observed by Sophia (in her reference frame) v is the relative velocity between the two observers (Sophia and Stella) c is the speed of light in a vacuum

Given: t = 1.90 hours (time observed by Sophia) v = 0.70c (velocity of Stella's spacecraft relative to Sophia, where c is the speed of light)

Plugging the values into the formula:

t' = 1.90 / sqrt(1 - (0.70c)^2/c^2) = 1.90 / sqrt(1 - 0.49) = 1.90 / sqrt(0.51) = 1.90 / 0.7141 ≈ 2.66 hours

Therefore, according to Sophia's clock, the movie would run for approximately 2.66 hours for Stella.

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