Elastic Collision Calculator

Understanding Elastic Collisions

In physics, a collision is called elastic if both momentum and kinetic energy are conserved. Unlike inelastic collisions, no energy is lost to heat, sound, or deformation. Elastic collisions are idealizations but can be closely observed in phenomena like gas molecules colliding or interactions between hard spheres such as billiard balls.

Key Principles

• Momentum Conservation: Total momentum before collision equals total momentum after collision.
• Kinetic Energy Conservation: No kinetic energy is lost during the collision.
• Relative Velocity: The relative speed of approach equals the relative speed of separation.

Formulas for One-Dimensional Elastic Collisions

For two masses m₁ and m₂ with initial velocities u₁ and u₂, the final velocities v₁ and v₂ are:

v₁ = ( (m₁ - m₂)u₁ + 2m₂u₂ ) / (m₁ + m₂)
v₂ = ( (m₂ - m₁)u₂ + 2m₁u₁ ) / (m₁ + m₂)

Worked Example

Suppose a 2 kg ball moving at 4 m/s collides elastically with a stationary 3 kg ball.

v₁ = ( (2-3)(4) + 2(3)(0) ) / (5) = (-4) / 5 = -0.8 m/s v₂ = ( (3-2)(0) + 2(2)(4) ) / (5) = 16 / 5 = 3.2 m/s

After the collision, the first ball moves at -0.8 m/s (reverses direction), and the second moves at 3.2 m/s. Momentum and kinetic energy remain conserved.

Applications

• Gas Laws: Kinetic theory of gases assumes perfectly elastic molecular collisions.
• Sports: Explains rebound of balls in tennis, billiards, and table tennis.
• Particle Physics: High-energy particle collisions often approximate elastic behavior.
• Engineering: Applied in impact mechanics and vibration analysis.
• Astrophysics: Used in modeling celestial body interactions under certain conditions.

Insights

While no real collision is perfectly elastic, many systems behave approximately elastic under certain conditions. Elastic collision equations are fundamental in physics education, statistical mechanics, and understanding how microscopic particle interactions lead to macroscopic properties like pressure and temperature.