You are a space traveler, moving at 0.60 c with respect to Earth, on your way to a star that is stationary relative to Earth. You measure the length of your trajectory to be 8.0 light-years (ly). Your friend makes the same journey at 0.80 c with respect to Earth. What does your friend measure the length of the trajectory to be? (Note: Light year is a unit of length)
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To find the length of the trajectory measured by your friend, we can use the concept of length contraction from the theory of relativity. The formula for length contraction is: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] Where: - \( L \) is the contracted length (what your friend measures). - \( L_0 \) is the proper length (the length of the trajectory you measured, 8.0 ly). - \( v \) is the speed of your friend (0.80c). - \( c \) is the speed of light. First, let's calculate the length contraction for your friend's speed: 1. Calculate \( v^2/c^2 \): \[ \frac{v^2}{c^2} = (0.80)^2 = 0.64 \] 2. Substitute this into the contraction formula: \[ L = 8.0 \sqrt{1 - 0.64} \] \[ L = 8.0 \sqrt{0.36} \] \[ L = 8.0 \times 0.6 \] \[ L = 4.8 \text{ ly} \] So, your friend measures the length of the trajectory to be 4.8 light-years.