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 solve this problem, we will use the concept of length contraction from the theory of special relativity. The length contraction formula is given by: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] where: - \(L\) is the contracted length (the length measured by the observer in motion), - \(L_0\) is the proper length (the length measured in the rest frame of the object), - \(v\) is the velocity of the moving observer, - \(c\) is the speed of light. ### Step 1: Identify known values - The proper length \(L_0\) (the length of the trajectory as measured by you) is 8.0 light-years. - Your speed \(v_1 = 0.60c\). - Your friend's speed \(v_2 = 0.80c\). ### Step 2: Calculate the length measured by your friend Your friend is moving at a speed of \(0.80c\) relative to Earth. To find the length of the trajectory as measured by your friend, we will use the length contraction formula. First, we need to calculate the contracted length \(L\) for your friend: \[ L = L_0 \sqrt{1 - \frac{v_2^2}{c^2}} \] Substituting the values: \[ L = 8.0 \, \text{ly} \cdot \sqrt{1 - (0.80)^2} \] Calculating \( (0.80)^2 \): \[ (0.80)^2 = 0.64 \] Now substituting this back into the equation: \[ L = 8.0 \, \text{ly} \cdot \sqrt{1 - 0.64} \] Calculating \(1 - 0.64\): \[ 1 - 0.64 = 0.36 \] Now taking the square root: \[ \sqrt{0.36} = 0.6 \] Finally, substituting this value back into the equation for \(L\): \[ L = 8.0 \, \text{ly} \cdot 0.6 \] Calculating the final length: \[ L = 4.8 \, \text{ly} \] ### Conclusion Your friend measures the length of the trajectory to be **4.8 light-years**.