Conservation of Energy — KE, PE, and Elastic Energy — The Complete Physics Guide
Energy Converter tracks kinetic and potential energy in real time as a ball rolls through ramps, springs, boosts, and friction pads — the live energy bars show every joule accounted for at every instant. This is the single most powerful idea in classical mechanics: energy is never created or destroyed, only converted between forms. Learning to track those conversions is what lets you solve motion problems that would be nearly impossible using forces and accelerations alone.
The Forms of Mechanical Energy
Kinetic energy KE = ½mv² is the energy an object has because it's moving. Gravitational potential energy PE = mgh is the energy an object has because of its height in a gravitational field. Elastic potential energy E_spring = ½kx² is the energy stored in a compressed or stretched spring, proportional to the square of the displacement from its natural length.
In a system where only conservative forces act — gravity and ideal springs, but not friction — total mechanical energy E = KE + PE is exactly constant. As a ball rolls down a ramp, PE converts to KE; as it rolls back up the other side, KE converts back to PE. The total never changes, only the split between the two forms. This is exactly what the live energy bars in this game are showing you frame by frame.
Friction breaks this symmetry. It's a non-conservative force — it always opposes motion and always removes mechanical energy from the system, converting it irreversibly into heat. Once energy has been converted to heat by friction, there's no ramp or spring configuration that gets it back into kinetic or potential form. This is why every level in Energy Converter that includes a friction pad is fundamentally about energy budgeting: you have to reach the goal with exactly the KE you need, knowing some has been permanently spent along the way.
The Work-Energy Theorem
The work-energy theorem connects forces directly to changes in motion: the net work done on an object equals its change in kinetic energy, W_net = ΔKE = ½mv² − ½mu². This falls directly out of Newton's Second Law combined with the SUVAT kinematics equations, and it's often far more useful in practice than working with F = ma directly, because it sidesteps the need to know exactly how the force varies over time.
For a conservative force, the work it does equals the negative change in potential energy: W = −ΔPE. Combined with the work-energy theorem, this gives ΔKE + ΔPE = 0 — total mechanical energy is conserved. For a non-conservative force like friction, the same combination gives ΔKE + ΔPE = W_friction, where W_friction is negative — mechanical energy decreases by exactly the (negative) work friction does.
This is also why braking distance scales with the square of speed, not speed itself. Kinetic energy KE = ½mv² must be entirely removed by the (roughly constant) friction force from the brakes, and since KE grows as v², so does the distance needed to remove it. Doubling your speed quadruples your stopping distance — a fact with serious real-world safety implications that follows directly from this one equation.
Energy in Springs
A spring compressed or stretched by distance x from its natural length stores elastic potential energy E = ½kx², where k is the spring constant (N/m) — a measure of stiffness. A spring with k = 400 N/m compressed by 0.05 m stores E = ½ × 400 × 0.05² = 0.5 J. When released against a ball of mass 0.1 kg on a frictionless surface, all of that energy converts to kinetic energy: 0.5 = ½ × 0.1 × v² → v = √10 ≈ 3.16 m/s.
Notice that spring energy depends on the square of compression, just like kinetic energy depends on the square of velocity — this parallel isn't a coincidence. Both a spring's restoring force and simple harmonic motion generally share the same underlying mathematics, and the boost pads in this game are a simplified stand-in for exactly this kind of stored, releasable energy.
Energy Exchange in Projectile Motion
When a ball leaves a ramp and becomes a projectile, its horizontal kinetic energy stays constant throughout the flight, since gravity — the only force acting — is purely vertical and does no work on horizontal motion. All the interesting energy exchange happens vertically: as the ball rises, vertical KE converts to PE; as it falls, PE converts back to vertical KE.
At the peak of its trajectory, vertical velocity is momentarily zero, so all of the initial vertical kinetic energy has become potential energy: ½mu_y² = mgh_max, giving the maximum height h_max = u_y²/(2g). This is exactly the same relationship you'd get from the SUVAT equation v² = u² − 2gh with v = 0 — energy methods and kinematics always agree, because one is derived from the other.
Worked Example — Ramp, Friction, and a Spring Launch
Problem: A 0.2 kg ball starts at rest at the top of a frictionless 2 m high ramp, then crosses 1.5 m of a friction pad with coefficient μ = 0.3 before reaching a spring with k = 250 N/m. How far is the spring compressed at maximum, assuming the ball loses all its remaining kinetic energy to the spring?
PE at top: mgh = 0.2 × 9.81 × 2 = 3.92 J
Energy lost to friction: W_f = μmg·d = 0.3 × 0.2 × 9.81 × 1.5 = 0.88 J
KE remaining at the spring: 3.92 − 0.88 = 3.04 J
Spring compression: ½kx² = 3.04 → x² = 2×3.04/250 = 0.0243 → x = 0.156 m (about 15.6 cm)
Real-World Applications
Roller coasters: The entire ride is a demonstration of energy conservation. The chain lift does work to raise the cars to the highest point, giving them maximum PE; every subsequent hill, loop, and drop is simply that stored energy converting back and forth into KE, with track friction and air resistance slowly bleeding energy away — which is why the first hill is always the tallest.
Regenerative braking: Electric and hybrid vehicles recover kinetic energy during braking by running the electric motor in reverse as a generator, converting some of the car's KE back into stored electrical energy instead of wasting it entirely as heat in the brake pads — a direct, deliberate exploitation of the work-energy theorem.
Pile drivers and forging hammers: Both work by converting gravitational PE (a raised weight) into kinetic energy during the fall, then transferring that KE almost instantaneously into work done on the target — driving a pile into the ground or shaping hot metal. The height the weight is raised to directly determines the KE, and therefore the force, available at impact.
Bungee jumping and trampolines: Both are close real-world analogues of the spring levels in this game. A stretched bungee cord or a trampoline's springs store elastic potential energy exactly like E = ½kx², and the entire thrill of the activity is the rapid, repeated conversion between that stored elastic energy and the kinetic and gravitational potential energy of the person moving through it.