Unit 4 - Problem 1 ==> Rollercoaster Design

Given

• A rollercoaster car with mass m at height h goes into a loop.
• At the top of the loop of radius r the car must go a velocity of V to be safe
• What is the minimum height the car must start at?

Rationale

• $$E_{potential} = E_{top-of-loop} + E_{kinetic-top-of-loop}$$
• $$m_{car} a_{gravity} h_x = (m_{car} a_{gravity} h_{top-of-loop}) + (\frac{1}{2} m_{car} V^2_{top-of-loop})$$
• $$a_{gravity} h_x = (a_{gravity} h_{top-of-loop}) + (\frac{1}{2} V^2_{top-of-loop})$$

Calculations

• $$10h_x = (10*20) + (\frac{1}{2}*10^2)$$
• $$10h_x = 200 + 50 = 250$$
• $$h_x = 25$$

Height vs Loop Radius

• Derivation from MIT Professor Lewin at about 28:40 on the video.
• At any point along the track, $$mgh = mgy + \frac{1}{2}m v^2$$
• Which simplifies to, $$2g (h-y) = V^2$$
• At the lop of the loop, $$y = 2r$$
• From previous work, at the apex of the loop,  $$a_{centripetal} = g = \frac{V^2}{R}$$
• $$V^2 = g * R$$
• Therefore, $$2g*(h-2r) \ge g*r$$
• $$2h - 4r \ge r$$
• $$h  \ge \frac{5}{2} r$$
• Regardless of mass of object, for it to stay on the track, it must originate from a height of at least 2.5 times the radius of the loop.