Design a roller coaster track and verify conservation of energy at every point. Calculate the minimum height needed to complete a loop.
Use the Surface tool to draw a roller coaster track — start with a high starting hill, a valley, and a circular loop. Click point by point to define the track shape. Set the track coefficient of friction to 0 initially (frictionless).
Place a Cart body on the starting hill and release it. Enable the graph panel and add two plots: KE (kinetic energy) and PE (potential energy) vs time. Observe: as KE rises, PE falls by exactly the same amount. Their sum — total mechanical energy — stays constant.
At the top of the loop, the centripetal force must equal or exceed gravity: mg = mv²/r → v_min = √(gr) at the top. Using energy conservation from the starting height h: v² = 2g(h – 2r). Combining: the minimum starting height is h_min = 2r + r/2 = 2.5r.
At the bottom of the loop, the normal force is highest. Enable the Contact force readout on the cart. At the bottom, N = mg + mv²/r — you'll see it spike as the cart passes through. At the top, N = mv²/r − mg (and can reach zero at minimum speed).
Set the track friction coefficient to μ = 0.02. Re-run and watch the energy graph: total mechanical energy now decreases over time, converted to heat. How high does the starting hill now need to be to still complete the loop? Adjust and find the new minimum height experimentally.
Discussion
Questions, corrections, and insights welcome.