💥Elastic Collisions
Collision Lab
Experiment with elastic and inelastic collisions. Adjust masses and velocities, then hit Launch. Watch how momentum is always conserved — but kinetic energy is only conserved in elastic collisions.
Perfectly Elastic Collisions
An elastic collision is one where both momentum AND kinetic energy are conserved.
In reality, perfectly elastic collisions are rare (billiard balls come close). But they're important conceptually and on the AP exam.
Two conservation equations: 1. Momentum: m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f 2. Kinetic Energy: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f²
These two equations let you solve for two unknowns (the two final velocities).
Special case: If m₁ = m₂ and m₂ is at rest, then the first object stops completely and the second moves at the first object's initial speed — a perfect velocity exchange!
✏️ Worked Example
Problem: A 2 kg ball moving at 6 m/s hits a stationary 2 kg ball. What are the final velocities? (elastic collision)
📐 Key Equations
Elastic Collisions
m_1v_1i + m_2v_2i = m_1v_1f + m_2v_2f(1)/(2)m_1v_1i² + (1)/(2)m_2v_2i² = (1)/(2)m_1v_1f² + (1)/(2)m_2v_2f²⚠️ Common Mistakes
Misconception: Thinking kinetic energy is always conserved in any collision.
✓ Correct thinking: KE is only conserved in elastic collisions. Most real collisions are inelastic — they lose KE to heat, sound, and deformation.
Why: Elastic means no permanent deformation and no energy converted to internal energy. Billiard balls are nearly elastic; car crashes are very inelastic.
Misconception: Forgetting to apply BOTH conservation laws when solving elastic collision problems.
✓ Correct thinking: Elastic collisions require two equations: conservation of momentum AND conservation of kinetic energy. You need both to find two unknowns.
Why: With only one equation you can't uniquely determine two final velocities. The energy equation provides the second constraint.
Misconception: Applying the "equal masses → velocity swap" rule to unequal-mass collisions.
✓ Correct thinking: The velocity exchange (first stops, second takes original speed) only applies when the two masses are equal AND one starts at rest.
Why: For unequal masses in elastic collisions, you must use both conservation equations to solve for each final velocity.
📝 Practice Problems
Try these problems. Check your answer when ready.
A 3 kg ball moving at 4 m/s hits a stationary 3 kg ball in an elastic collision. What are the final velocities of each ball?
Is the following collision elastic? Ball A (2 kg) at 6 m/s hits stationary Ball B (2 kg). After: A at 2 m/s, B at 4 m/s.
A 4 kg ball at 10 m/s elastically collides with a stationary 1 kg ball. Find both final velocities.
v_1f = (m_1 - m_2)/(m_1 + m_2)v_1i, v_2f = (2m_1)/(m_1 + m_2)v_1iAfter an elastic collision between a 5 kg object (initially at 8 m/s) and a stationary 3 kg object, the 5 kg object moves at 2 m/s. Find the final speed of the 3 kg object and verify KE conservation.
A neutron (mass m) moving at speed v₀ undergoes an elastic head-on collision with a stationary carbon nucleus (mass 12m). What fraction of the neutron's KE is transferred to the carbon nucleus?
v_2f = (2m)/(m + 12m)v_0 = (2)/(13)v_0Two balls undergo an elastic collision. Ball 1 (1 kg) moves at +6 m/s; Ball 2 (2 kg) moves at −3 m/s (toward each other). Find both final velocities.
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