# Brachistochrone Problem Solution The Brachistochrone problem is about finding the path of quickest descent between two points under gravity, which is a cycloid. ## 1. Cycloid Parameters The cycloid curve can be described parametrically as: \[ x = r(\phi - \sin\phi) \] \[ y = r(1 - \cos\phi) \] ### Determine Parameters \(\phi_B\) and \(r\) Given: - \(A = (, )\) - \(B = (2. \, \text{m}, 1. \, \text{m})\) We need to solve for \(\phi_B\) and \(r\) such that: \[ 2. = r(\phi_B - \sin\phi_B) \] \[ 1. = r(1 - \cos\phi_B) \] Divide the equations to eliminate \(r\): \[ \frac{2.}{1.} = \frac{\phi_B - \sin\phi_B}{1 - \cos\phi_B} \] This is a transcendental equation that can be solved numerically for \(\phi_B\). After finding \(\phi_B\), use either equation to solve for \(r\). ## 2. Compute Travel Time \(T_{\text{cycloid}}\) The travel time \(T\) for a cycloid is given by: \[ T = \frac{\phi_B}{\sqrt{g/r}} \] Substitute the values of \(\phi_B\), \(r\), and \(g = 9.81 \, \text{m/s}^2\) to find \(T_{\text{cycloid}}\). ## 3. Comparison with Other Paths ### Straight-Line Ramp The straight-line path from \(A\) to \(B\) is the direct line. Compute its angle \(\theta\) and use: \[ v = \sqrt{2g h} \] where \(h\) is the vertical drop, to find the time down the ramp. ### Vertical-then-Horizontal Path 1. **Vertical Drop**: \(y = 1 \, \text{m}\) down. 2. **Horizontal Coast**: \(x = 2 \, \text{m}\) horizontally at velocity from the drop. Compute the times for these two segments and sum them. ## Summary 1. Solve for \(\phi_B\) and \(r\) numerically. 2. Calculate \(T_{\text{cycloid}}\). 3. Compare with times for other paths for decision-making. This approach provides a clear understanding of the fastest path under gravity in a frictionless environment.