Conservation of Energy
The law of conservation of energy states that the total energy of an isolated system remains constant over time: energy can neither be created nor destroyed, only transformed from one form to another. In mechanical systems, this means the sum of kinetic energy and potential energy remains constant in the absence of non-conservative forces such as friction. This principle, one of the most fundamental in all of science, is derived mathematically from Noether's theorem as a consequence of the time-translation symmetry of physical laws.
Key Formula
½mv_i² + mgh_i = ½mv_f² + mgh_f
LaTeX: KE_i + PE_i = KE_f + PE_f
| Symbol | Meaning | Unit |
|---|---|---|
| KE_i | Initial kinetic energy | J |
| PE_i | Initial potential energy | J |
| KE_f | Final kinetic energy | J |
| PE_f | Final potential energy | J |
| v_i, v_f | Initial and final speeds | m/s |
| h_i, h_f | Initial and final heights | m |
Worked Example
Problem
A 3 kg ball is dropped from a height of 5 m. Using conservation of energy, find its speed just before it hits the ground. (Ignore air resistance; g = 9.8 m/s²)
Solution
Step 1 — Set reference: Take the ground as h = 0 (PE = 0 at ground). Step 2 — Initial state: Ball at rest at h = 5 m. KE_i = 0 J; PE_i = mgh_i = 3 × 9.8 × 5 = 147 J. Total energy = 147 J. Step 3 — Final state: Ball at ground, h_f = 0. PE_f = 0. All energy is kinetic: KE_f = 147 J. Step 4 — Find speed: ½mv_f² = 147 → v_f² = (2 × 147) / 3 = 98 → v_f = √98 ≈ 9.9 m/s.
Answer
The ball hits the ground at approximately 9.9 m/s.
Energy distribution of a 3 kg ball (m = 3 kg, total energy = 147 J) falling from 5 m
| Height h (m) | GPE (J) | KE (J) | Total Energy (J) | Speed (m/s) |
|---|---|---|---|---|
| 5.0 (start) | 147.0 | 0.0 | 147.0 | 0.0 |
| 4.0 | 117.6 | 29.4 | 147.0 | 4.4 |
| 3.0 | 88.2 | 58.8 | 147.0 | 6.3 |
| 2.0 | 58.8 | 88.2 | 147.0 | 7.7 |
| 1.0 | 29.4 | 117.6 | 147.0 | 8.9 |
| 0.0 (ground) | 0.0 | 147.0 | 147.0 | 9.9 |
Interactive Tools
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Related Terms
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its state of motion. It depends on both the mass of the object and the square of its speed, meaning that doubling the speed quadruples the kinetic energy. Kinetic energy is transferred to objects through work and is a key quantity in collision analysis, transport safety, and the work-energy theorem.
Gravitational Potential Energy
Gravitational potential energy (GPE) is the energy stored in an object due to its height above a chosen reference level in a gravitational field. It increases with both the mass of the object and its height above the reference, and is fully convertible to kinetic energy as the object falls. GPE is fundamental to the analysis of projectiles, hydroelectric power generation, and the orbital mechanics of satellites.
Elastic Potential Energy
Elastic potential energy is the energy stored in a deformed elastic object — such as a stretched spring, a compressed rubber band, or a bent bow — that can be fully recovered when the deforming force is removed. It arises from the intermolecular forces within the elastic material that resist deformation and is equal to the work done in stretching or compressing the object. Elastic PE is pivotal in springs, archery, vehicle suspensions, and the molecular-scale understanding of elasticity.
The principle was articulated in its modern form by Julius Robert von Mayer (1842), James Prescott Joule (1843), and Hermann von Helmholtz (1847), who published "Über die Erhaltung der Kraft" (On the Conservation of Force). The deep reason for energy conservation — time-translation symmetry — was proven by Emmy Noether in 1915 (Noether's theorem). "Conservation" from Latin "conservatio" (a keeping, preserving), from "conservare" (to keep safe).