Collisions

How the model works

In a one-dimensional collision with no outside forces, total momentum is conserved:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

The coefficient of restitution e compares how fast the carts separate with how fast they approached: v₂ − v₁ = −e(u₂ − u₁). Solving both equations gives

v₁ = (m₁u₁ + m₂u₂ + m₂e(u₂ − u₁)) / (m₁ + m₂)
v₂ = (m₁u₁ + m₂u₂ + m₁e(u₁ − u₂)) / (m₁ + m₂)

• e = 1, elastic: kinetic energy is conserved.
• 0 < e < 1, inelastic: some kinetic energy becomes heat, sound and deformation.
• e = 0, perfectly inelastic: the carts stick together and lose the most kinetic energy that momentum conservation allows.

Velocities are positive to the right. The centre-of-mass velocity (m₁u₁ + m₂u₂)/(m₁ + m₂) never changes.

Ideal carts: instantaneous collision, no friction, no rotation. The track ends are walls that reflect carts elastically.

Guided experiments

1. Newton's cradle swap. Equal masses, elastic, cart 2 at rest. Prediction: cart 1 stops dead and cart 2 leaves with cart 1's original velocity. Equal-mass elastic
2. Sticky collision. Set e = 0 with equal masses, one moving at 2 m/s. Prediction: they move off together at 1 m/s. Momentum is unchanged but half the kinetic energy is gone. Perfectly inelastic
3. Ping-pong ball hits a bowling ball. m₁ = 0.5 kg moving at 3 m/s hits m₂ = 10 kg at rest, elastic. Prediction: the light cart bounces back at almost 3 m/s, while the heavy one barely moves (about 0.29 m/s). Light hits heavy

Challenge

A 2 kg cart moving at 4 m/s hits a 2 kg cart at rest. Afterwards cart 2 moves at 3 m/s. What is e, and what fraction of kinetic energy was lost?

References

• OpenStax, University Physics Volume 1, §9.4 “Types of Collisions”.
• HyperPhysics, “Coefficient of restitution”.