A rod of proper length \(10 \mathrm{~cm}\) moves longitudinally along the \(x\)-axis of S at speed \(\frac{1}{2} c\). How long (in S) does it take a particle, moving oppositely at the same speed, to pass the rod?

According to the length contraction formula in special relativity, the length of an object moving at a relativistic speed will appear to be shorter in the observer's frame. The contracted length (L') of the rod can be calculated by the following formula: \(L' = L\sqrt{1 - \frac{v^2}{c^2}}\) Where: - L is the proper length of the rod, which is 10 cm - v is the speed of the rod, which is 0.5c - c is the speed of light Plugging in the values and calculating L': \(L' = 10\sqrt{1 - \frac{(0.5c)^2}{c^2}} = 10\sqrt{1 - 0.25} = 10\sqrt{0.75} \approx 8.66 \mathrm{~cm}\)

As both the rod and the particle are moving along the x-axis, we will apply the Lorentz transformation for the x-component of position: \(x = \gamma(x' + vt')\) Where: - x is the position in the observer's frame (S) - x' is the position in the moving frame (S') in which the rod is at rest - v is the relative velocity between the frames (S and S') - t' is the time in the moving frame (S') - γ (gamma) is the Lorentz factor, given by the relation: ℝ^{1/2} Since the particle is moving oppositely at the same speed and approaching the rod, the relative velocity between the particle and the rod is (0.5c - (-0.5c)) = c. Now, let's find γ at v=c: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - 1}}\) Unfortunately, calculating the Lorentz factor for v=c results in division by zero, which is not allowed. Therefore, the Lorentz transformation cannot be applied in this scenario.

Since the Lorentz transformation cannot be applied in this case, it is an indication that such a scenario is not physically possible in the context of special relativity. Objects with mass (rod and particle in this case) cannot reach the speed of light, as it would require infinite energy. So, it is not possible to calculate the time required for the particle to pass the rod. However, for objects moving at speeds very close to the speed of light (but not exactly c), the Lorentz transformations can still be applied, and length contraction and time dilation will occur accordingly.