Roller Coaster Energy Transfer (2D)

This 2D companion to the 3D Roller Coaster Energy Transfer simulation is built around one exact identity: E₀ = KE + PE + Heat, checked and redrawn every frame instead of merely implied by the animation. Pick one of four spline track profiles — a classic hill-and-loop, a no-loop double hump, a steep drop into a large loop, or a gentle always-clears valley — then use the hill-height scale, start height, rolling-friction, air-drag and cart-mass sliders to change how much energy the cart carries and how quickly it loses it. The stacked energy bar always fills to exactly E₀ (kinetic energy in red, potential energy in blue, heat lost to friction and drag in grey), a scrolling energy-vs-time chart shows the same three quantities trading off continuously, and a live physics check reports either the loop's contact margin v²/(g·r) or how much height margin the cart has over the next hill's peak — backed by the analytically derived minimum launch height for each track. Starve the cart of energy and it either loses contact at the top of a loop and falls, or stalls short of a hill's peak and rolls back to oscillate and settle in the valley below, exactly as conservation of energy predicts.