Potential Energy Curves and Turning Points

Plot potential energy U against position x, then draw a horizontal line at the total mechanical energy E. Since E = U + K, the gap between that line and the curve at any point is the kinetic energy there. Where the line sits above the curve, K is positive and the object can be. Where the curve rises above the line, K would have to be negative, which is impossible, so the object can never reach there. The points where the line crosses the curve are turning points: K = 0, the object momentarily stops, and the force reverses its motion.

The slope of the U curve gives the force directly: F = -dU/dx. Where the curve falls as x increases, the slope is negative, so the force is positive, pushing the object toward lower U. Potential energy curves are really just hills in disguise: objects always get pushed downhill on the U-vs-x graph, same as a ball rolls downhill on an actual hill. That single idea replaces separate force diagrams at every position with one picture of the whole journey.

Flat spots on the curve, where slope is zero, are equilibrium points. A dip (local minimum) is stable: nudge the object either way and the slope points back toward the bottom, restoring it. A peak (local maximum) is unstable: nudge it either way and the slope pushes it further away. This connects directly to the energy bar charts from earlier in the unit, since a turning point is just the instant a kinetic energy bar hits zero, and it sets up power and later unit topics where the rate of energy change, not just its amount, starts to matter.