## Solution to the Brachistochrone Problem To solve the Brachistochrone problem, we need to find the path of a bead under gravity that results in the shortest travel time from point \( A = (, ) \) to point \( B = (2. \, \text{m}, 1. \, \text{m}) \). ### 1. Find the Cycloid A cycloid is the curve of fastest descent under gravity. The parametric equations for a cycloid are: \[ x = r (\phi - \sin \phi) \] \[ y = r (1 - \cos \phi) \] Where: - \( r \) is the generating radius. - \( \phi \) is the parameter. To find the cycloid that passes through \( B = (2. \, \text{m}, 1. \, \text{m}) \), we solve these equations for \( r \) and \( \phi_B \): 1. \( r (\phi_B - \sin \phi_B) = 2. \) 2. \( r (1 - \cos \phi_B) = 1. \) From equation (2), solve for \( r \): \[ r = \frac{1.}{1 - \cos \phi_B} \] Substitute \( r \) into equation (1): \[ \frac{1.}{1 - \cos \phi_B} (\phi_B - \sin \phi_B) = 2. \] Solve this equation to find \( \phi_B \) and subsequently \( r \). ### 2. Compute the Travel Time \( T_{\text{cycloid}} \) The travel time along the cycloid is given by: \[ T_{\text{cycloid}} = \frac{\phi_B}{\sqrt{g/r}} \] Where \( g = 9.81 \, \text{m/s}^2 \). ### 3. Compare with Other Paths #### a. Straight-Line Ramp The straight-line path from \( A \) to \( B \) can be calculated using basic physics equations for constant acceleration. #### b. Vertical-then-Horizontal Path 1. Drop directly to \( y = 1. \, \text{m} \). 2. Travel horizontally to \( x = 2. \, \text{m} \). Calculate the time for each segment and sum them. ### Conclusion Calculate \( T_{\text{cycloid}} \), \( T_{\text{ramp}} \), and \( T_{\text{vertical-horizontal}} \) to compare the times and verify that the cycloid provides the shortest travel time.

# Brachistochrone Problem Overview The Brachistochrone problem investigates the quickest path for a bead sliding from point \( A \) to point \( B \) under the influence of gravity. Here, we are tasked with computing this path in a vertical plane. ## Problem Parameters - **Start Point**: \( A = (0, 0) \) - **End Point**: \( B = (2.0 \, \text{m}, 1.0 \, \text{m}) \) - **Gravity**: \( g = 9.81 \, \text{m/s}^2 \) ## 1. Finding the Cycloid ### Cycloid Equations The cycloid can be described with the following parametric equations: \[ x = r (\phi - \sin \phi) \] \[ y = r (1 - \cos \phi) \] ### Determine \( r \) and \( \phi_B \) To find the parameters for the cycloid that connects points \( A \) and \( B \), we need to solve the following equations: 1. \( r (\phi_B - \sin \phi_B) = 2 \) 2. \( r (1 - \cos \phi_B) = 1 \) #### Solving for \( r \) From the second equation: \[ r = \frac{1}{1 - \cos \phi_B} \] #### Substitute into the First Equation Plug \( r \) into the first equation: \[ \frac{1}{1 - \cos \phi_B} (\phi_B - \sin \phi_B) = 2 \] This equation can be solved numerically for \( \phi_B \), giving us both \( \phi_B \) and \( r \). ## 2. Compute the Travel Time \( T_{\text{cycloid}} \) The travel time along the cycloid is given by the formula: \[ T_{\text{cycloid}} = \frac{\phi_B}{\sqrt{g/r}} \] Where \( g = 9.81 \, \text{m/s}^2 \) and \( r \) is determined from the previous calculations. ## 3. Compare with Other Paths ### a. Straight-Line Ramp For the straight-line path, calculate the distance and the time using: \[ \text{Distance} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] This path experiences uniform acceleration due to gravity. ### b. Vertical-then-Horizontal Path 1. **Vertical Drop**: The bead drops vertically from \( y = 0 \) to \( y = 1 \, \text{m} \). 2. **Horizontal Travel**: The bead then moves horizontally from \( x = 0 \) to \( x = 2 \, \text{m} \). #### Time Calculation - **Vertical Drop Time**: Use \( t = \sqrt{\frac{2h}{g}} \) - **Horizontal Travel Time**: Use \( t = \frac{d}{v} \) where \( v \) is the speed after the vertical drop. ### Conclusion By calculating \( T_{\text{cycloid}} \), \( T_{\text{ramp}} \), and \( T_{\text{vertical-horizontal}} \), we can demonstrate that the cycloid provides the shortest travel time among the considered paths.