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)

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.