Types of Collisions

In a perfectly elastic collision, both momentum AND kinetic energy are conserved. This means the total KE before equals the total KE after: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f². Real-world examples that come close to elastic include collisions between hard steel balls, atomic or subatomic particle collisions, and idealized billiard ball impacts. In elastic collisions between two objects, we can derive specific formulas for the final velocities using both conservation laws simultaneously.

In an inelastic collision, momentum is conserved but kinetic energy is NOT conserved — some KE is lost to thermal energy, sound, or permanent deformation. Most real-world collisions are inelastic. A car fender crumpling in a crash is a deliberate application of inelastic collision engineering: the crumple zone absorbs kinetic energy, reducing the force on passengers. The kinetic energy 'lost' doesn't vanish — it transforms into other energy forms, consistent with the conservation of total energy.

A perfectly inelastic collision is a special case where the two objects stick together after impact and move as one combined mass. This maximizes kinetic energy loss while still conserving momentum. The conservation of momentum gives: m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)vf. Notice there is now only one final velocity since the objects move together. Examples include a ball of clay hitting a wall and sticking, a football tackle where players grab onto each other, or coupling train cars.

Step 1: Always apply conservation of momentum — it holds for all collision types. Step 2: Calculate the total kinetic energy before and after the collision. If KE_total is the same before and after → elastic. If KE_total decreases and the objects separate → inelastic. If the objects stick together → perfectly inelastic. A useful check: for elastic collisions in 1D, the relative speed of approach equals the relative speed of separation: (v₁ᵢ − v₂ᵢ) = −(v₁f − v₂f). This is sometimes called the 'coefficient of restitution = 1' condition.

• m_1, m_2 — Masses of objects 1 and 2 (kg): The masses of the two colliding objects, which remain constant throughout the collision.
• v_{1i}, v_{2i} — Initial velocities (m/s): The velocities of objects 1 and 2 before the collision. Direction matters — assign positive and negative signs carefully.
• v_{1f}, v_{2f} — Final velocities (m/s): The velocities of objects 1 and 2 after the collision. In a perfectly inelastic collision, v_{1f} = v_{2f} = v_f.

Problem. A 2.0 kg cart moving at 4.0 m/s to the right collides with a 3.0 kg cart that is stationary. After the collision, the 2.0 kg cart moves at 0.80 m/s to the right. (a) Find the velocity of the 3.0 kg cart after the collision. (b) Determine whether this collision is elastic or inelastic by comparing kinetic energies.

Answer. The 3.0 kg cart moves at approximately 2.1 m/s to the right. The collision is inelastic because kinetic energy decreased from 16.0 J to approximately 7.5 J.

• Conservation of Momentum — Momentum conservation is the foundational law that applies to ALL collisions — elastic and inelastic alike. Understanding it is prerequisite to analyzing any collision problem.
• Impulse and the Impulse-Momentum Theorem — Impulse describes the change in momentum during a collision. Understanding how force and time interact during a collision connects directly to why different collisions produce different energy outcomes.
• Kinetic Energy — Kinetic energy (½mv²) is the quantity that distinguishes elastic from inelastic collisions. You must be fluent with KE calculations to classify any collision correctly.
• Work-Energy Theorem — In inelastic collisions, kinetic energy is 'lost' — but really converted into other forms. The work-energy theorem explains how energy transforms between mechanical and thermal/deformation energy.