2. Two spaceships A and B are exploring a new planet. Relative to this planet, spaceship A has a speed of 0.60 c , and spaceship B has a speed of 0.80c. What is the ratio \( \mathrm{D}_{N} / \mathrm{D}_{0} \) of the values for the planet's diameter that each spaceship measures in a direction that is parallel to its motion?

To solve this problem, we need to consider the concept of length contraction in 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, - \( L_0 \) is the proper length (the length measured at rest), - \( v \) is the velocity of the observer relative to the object, - \( c \) is the speed of light. Given that spaceship A has a speed of 0.60c and spaceship B has a speed of 0.80c, we can calculate the contracted lengths for the planet's diameter as measured by each spaceship. Let's denote the diameter of the planet as \( D_0 \). The contracted diameter as measured by spaceship A is \( D_A \) and by spaceship B is \( D_B \). For spaceship A: \[ D_A = D_0 \sqrt{1 - \frac{(0.60c)^2}{c^2}} \] For spaceship B: \[ D_B = D_0 \sqrt{1 - \frac{(0.80c)^2}{c^2}} \] Now, we can calculate the ratio \( \frac{D_A}{D_B} \) by dividing the contracted diameters measured by each spaceship. The ratio of the values for the planet's diameter that each spaceship measures in a direction parallel to its motion is \( \frac{D_A}{D_B} = \frac{4}{3} \) or \( 1\frac{1}{3} \) or \( 1.\dot{3} \).