Elastic Collision
An elastic collision is one in which both the total kinetic energy and the total momentum of the system are conserved before and after the collision. No energy is lost to deformation, heat, or sound, making it an idealized model most closely approximated by atomic and subatomic particle interactions. Billiard ball collisions and gas molecule interactions are common approximations of elastic collisions.
Key Formula
½m1v1i² + ½m2v2i² = ½m1v1f² + ½m2v2f²
LaTeX: \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2
| Symbol | Meaning | Unit |
|---|---|---|
| m₁, m₂ | Masses of the two objects | Kilogram (kg) |
| v₁ᵢ, v₂ᵢ | Initial velocities | m/s |
| v₁f, v₂f | Final velocities after collision | m/s |
Worked Example
Problem
A 2 kg ball moving at 8 m/s collides elastically with a stationary 2 kg ball. Find the final velocities of both balls.
Solution
Step 1: For equal-mass elastic collisions, velocities are exchanged. Using the standard elastic collision formulas: v₁f = ((m₁ - m₂)/(m₁ + m₂))v₁ᵢ = ((2-2)/(2+2)) × 8 = 0 m/s Step 2: Calculate v₂f. v₂f = (2m₁/(m₁ + m₂))v₁ᵢ = (2×2/(2+2)) × 8 = (4/4) × 8 = 8 m/s Step 3: Verify momentum: 2×8 + 2×0 = 2×0 + 2×8 = 16 kg·m/s ✓ Verify KE: ½×2×64 = ½×2×64 = 64 J ✓
Answer
Ball 1 stops (v₁f = 0 m/s); Ball 2 moves at 8 m/s in original direction.
Elastic vs. Inelastic Collision Comparison
| Property | Elastic | Inelastic | Perfectly Inelastic |
|---|---|---|---|
| Momentum conserved | Yes | Yes | Yes |
| Kinetic energy conserved | Yes | No (partially lost) | No (maximum loss) |
| Objects separate? | Yes | Yes | No — they stick together |
| Real-world example | Billiard balls | Car crash | Clay balls merging |
| Coefficient of restitution | 1.0 | 0 < e < 1 | 0 |
Interactive Tools
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Related Terms
Inelastic Collision
An inelastic collision is one in which the total kinetic energy of the system is not conserved, though the total momentum remains conserved. The lost kinetic energy is converted into other forms such as heat, sound, or deformation energy. When two objects collide and stick together, it is termed a perfectly inelastic collision, representing the maximum possible loss of kinetic energy.
Conservation of Momentum
The law of conservation of momentum states that the total momentum of a closed, isolated system remains constant if no external net force acts on it. This means the sum of momenta of all objects before an interaction equals the sum after the interaction. It is one of the most fundamental conservation laws in physics and applies equally to collisions, explosions, and all mechanical interactions.
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.
From Greek "elastikos" meaning "springy" or "able to return to its original form", from "elaunein" (to drive). The term reflects the property of objects that rebound without permanent deformation.