Energy Conservation Simulator
Watch a ball roll along a track with hills. See kinetic and potential energy exchange in real time and verify total mechanical energy conservation.
About Energy Conservation
The Law of Conservation of Mechanical Energy states that in the absence of non-conservative forces (such as friction or air resistance), the total mechanical energy of a system remains constant. This total is the sum of kinetic energy (KE) and gravitational potential energy (PE).
Key Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| m | Mass | kg | Mass of the rolling ball |
| h₀ | Initial Height | m | Starting height of the ball |
| h | Current Height | m | Height of the ball at position x on track |
| g | Gravitational acceleration | m/s² | g = 9.81 m/s² |
| PE | Potential Energy | J | PE = mgh |
| KE | Kinetic Energy | J | KE = ½mv² |
| E | Total Mechanical Energy | J | E = KE + PE = mgh₀ (conserved) |
| v | Speed | m/s | v = √(2·KE/m) = √(2g(h₀ − h)) |
Worked Example
A 1 kg ball starts at height h₀ = 3 m with zero velocity on a frictionless track. Find its speed at h = 1 m.
Track Shape
The simulator uses a sinusoidal hill: h(x) = h₀ · sin(πx/L) for x ∈ [0, L]. The ball starts at one end, rolls over the hill, and the energy bars update continuously.
Effect of Mass on Energy
- Larger mass means more total energy (E = mgh₀), but the speed at any height is the same: v = √(2g(h₀ − h)).
- Mass cancels out in the speed formula — a heavier ball falls at the same rate as a lighter one (Galileo's insight).
- Mass does affect KE and PE magnitudes: doubling mass doubles both KE and PE at every point.
Energy Conservation Formulas
Gravitational Potential Energy
Kinetic Energy
Conservation of Mechanical Energy
Speed from Height
Track Height Profile
| Formula | Description | Notes |
|---|---|---|
| PE = mgh | Gravitational PE | Reference: track baseline (h = 0) |
| KE = ½mv² | Kinetic energy | Translational KE only (no rolling inertia) |
| E = KE + PE | Total mechanical energy | Conserved when friction = 0 |
| v = √(2g(h₀−h)) | Speed from energy | Mass cancels — speed independent of mass |
| E = mgh₀ | Initial total energy | All PE at start (v₀ = 0) |
Frequently Asked Questions
Why does mass affect total energy but not speed?
Total energy E = mgh₀ scales with mass, but when you solve for speed from KE = ½mv² = mgh₀ − mgh, the mass m cancels: v = √(2g(h₀−h)). So a 5 kg ball and a 0.1 kg ball reach the bottom at the same speed on the same frictionless track.
What happens to energy if friction is present?
Friction converts mechanical energy to heat. Total mechanical energy (KE + PE) decreases over time. The ball slows down more than energy conservation predicts. The simulator shows the frictionless ideal case.
Why does the ball slow down near the peak?
At the peak, the ball has maximum PE. Since total energy is fixed, KE is minimum at the top — meaning minimum speed. The ball speeds up again as it descends.
Can the ball make it over the hill if it starts lower?
Only if the initial height is at least as high as the hill peak. If h₀ < h_peak, the ball lacks sufficient energy to reach the top and would stop and reverse in a real system. The simulator's track height equals h₀, so the ball just barely reaches the top.
Is rotational kinetic energy included?
No. The simulator models translational KE only (½mv²). For a rolling ball, you'd also need ½Iω² = ¼mv² for a solid sphere, reducing the translational speed by a factor of √(5/7). This simulator assumes a sliding point mass for clarity.