Conservation of Mechanical Energy

Instructions for Students & Teachers

Use the simulator above to explore conservation of mechanical energy. This simulator tracks Kinetic Energy, Gravitational Potential Energy, and (when friction is on) energy lost to friction, so you can watch energy convert between forms — or leak out of the system — as a ball moves along a track.

How to Use the Conservation of Energy Simulator

View
Use the Dark Mode toggle to switch the graph between light and dark themes.

Track Shape
Choose a preset track: Bowl, Double Hump, Loop, or S-Curve. Each one gives students a different shape to reason through — a Double Hump is a good one for asking whether the ball can clear the second hump, and a Loop raises the question of minimum speed at the top.
Edit Track lets you drag the individual control points along the curve to reshape the track by hand, instead of using a preset.
Note for teachers: Edit Track is worth using if you want students to design a track that satisfies a constraint themselves (for example, "build a track where the ball just barely makes it over both humps") rather than just observing a track you built for them.

Parameters
Mass, Start Height, and Initial Speed set the ball's starting conditions.
Gravity has a numeric field plus four one-click presets: Earth 9.8, Earth (AP) 10, Moon 1.6, and Jupiter 24.8 — useful for quickly comparing how the same track behaves under different gravity without retyping the value.
Friction toggles whether the track removes mechanical energy as the ball moves. With friction off, total mechanical energy stays constant; with it on, the energy bar chart will show a nonzero Wf (work done by friction) by the end of the run.

Display
Scale Grid, h=0 Reference Line, Show Ball Labels, and Velocity Vector each toggle that element on or off in the graph above.
The h=0 Reference field lets you set exactly where "zero height" is for the purposes of calculating gravitational PE — you can also just drag the dashed reference line directly on the graph.
Note for teachers: dragging the h=0 line to a new position and watching Ug recalculate in real time is a good way to show students that gravitational PE is always relative to a chosen reference point, not an absolute quantity.

Simulation
Playback Speed: ¼×, ½×, 1×, 2×, or 4×.
Play runs the simulation from the current start height and speed; Reset returns the ball to its starting position.
Live Values shows the ball's current height, Kinetic Energy, Gravitational PE, and total mechanical energy, updating as the simulation runs.
Below the track, the Energy bar chart breaks down K, Ug, Em, and Wf at the current moment, and the Energy vs. Time graph (fills in once you run the simulation) plots how each of those quantities changes over the course of the run.
You can also drag the ball directly to a new starting position on the track before hitting Play.

Challenge Mode (Test your understanding!)
Difficulty: Easy, Medium, or Hard.
Easy: no friction, and you're asked to find one energy value (like Kinetic Energy) at a given height.
Medium: adds more moving pieces to solve for between the given states.
Hard: friction is involved, and you're typically asked to find the energy removed by friction (Wf) between two states described by their heights and speeds.
Mastery tip: on Hard challenges, start by calculating total mechanical energy at the state you know completely, then use conservation (minus friction losses) to work toward what's missing — don't try to guess the answer directly from the numbers given.
Click New Challenge for a new problem at the current difficulty. Enter your answer and click Check.