Problem 1418


Determine the height h to the top of the incline D to which the 200 kg roller coaster car will reach, if it is launched at B with a speed just sufficient for it to round the top of the loop at C without leaving the track. The radius of curvature at C is r_{C} = 25m. To see the system in motion press here


First, determine the velocity of the car at the top of the loop at C necessary to maintain contact with the tracks. Drawing a free body diagram of the car at the top of the loop at C. The equation of motion in the normal direction gives (neglecting the normal force) (1) mg = mv^{2}/r Which gives v_{C} = (rg)^{1/2} = 15.66 m/s Now, we find the required initial velocity at B, using conservation of energy. T_{B} + V_{B} = T_{C} + V_{C} (2) 1/2 m v_{B}^{2} + 0 = 1/2 m v_{C}^{2 }+ mg h_{C} Substituting the values and solving the above equation gives v_{B} = 30.53 m/s Now, to find the maximum height at D, we again apply conservation of energy. T_{B} + V_{B} = T_{D} + V_{D} (3) 1/2 m v_{B}^{2} + 0 = m g h_{D} Solving equation (3) gives the maximum height as. h_{D} = 47.5 m


In the ADAMS simulation, the roller coaster is given the initial velocity of 30.53 m/s as calculated analytically. The height, speed, and kinetic energy of the roller coaster are shown. Notice that 1. the velocity at the top of the loop at C is v_{C} = 15.5 m/s. 2. The maximum height at D, corresponding to the point where the velocity is zero is h = 47.5m. 3. The kinetic energy is the opposite of the height. Since the only potential energy in the system is due to gravity, this reflects the principle of conservation of energy.
