Elastic & Inelastic Collisions

1. Definitions (Must Know)

A. Collision (impulsive interaction)

A collision is a short interaction where two bodies exert large forces on each other for a short time, so their momenta change significantly.

B. Elastic collision

An elastic collision is one where:

• total momentum is conserved, and
• total kinetic energy is conserved.

C. Inelastic collision

An inelastic collision is one where:

• total momentum is conserved, but
• total kinetic energy is not conserved (some is transferred to other energy stores, e.g. heating, sound, deformation).

D. Completely (perfectly) inelastic collision

A completely inelastic collision is an inelastic collision where the objects stick together after impact, so they have the same final velocity.

E. Impulse

Impulse is the change in momentum:

vector J = Δ vector p

See: Momentum & Impulse.

2. Key Ideas (What Earns Marks)

• Treat 1D collisions with a sign convention (e.g. right is +).
• Momentum conservation is a system law: ∑ p_before = ∑ p_after provided external impulse is negligible.
• Kinetic energy is conserved only in elastic collisions.
• Completely inelastic: “stick together” ⇒ v₁ = v₂ after collision.
• Perfectly elastic (1D): speed of approach = speed of separation: |u₁-u₂| = |v₂-v₁|
• Average collision force comes from impulse: F_avg = (Δ p)/(Δ t)

3. Detailed Explanations

A. When can you conserve momentum?

Momentum is conserved for a system when the resultant external force is zero, or when the external impulse is negligible during the short collision time.

This is why collision questions often say “smooth surface” or “no external forces”.

B. 1D collision setup (exam workflow)

1. Choose a positive direction.
2. Label initial velocities as u₁, u₂ and final velocities as v₁, v₂ (include signs).
3. Apply momentum conservation: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
4. Use the collision type to get a second equation (if needed).

C. Completely inelastic collisions (stick together)

If the objects stick, they move together with a common velocity v:

m₁u₁ + m₂u₂ = (m₁ + m₂)v

Kinetic energy decreases because energy is transferred to other stores during deformation.

D. Perfectly elastic collisions (1D shortcut)

For a perfectly elastic collision in 1D, you use:

• momentum conservation, and
• speed of approach = speed of separation: |u₁-u₂| = |v₂-v₁|

This is equivalent to the “relative speed of approach equals relative speed of separation” statement in the syllabus.

E. Impulse and force in collisions

Collision forces vary with time, so questions often ask for an average force:

F_avg = (Δ p)/(Δ t)

For the two colliding objects, Newton’s 3rd law implies the impulses are equal in magnitude and opposite in direction.

4. Common Mistakes

• Forgetting momentum is a vector (dropping signs in 1D).
• Conserving momentum for one object instead of the system.
• Assuming “collision = elastic” (elastic must be stated or proven with KE/relative-speed rule).
• Using N = mg or friction assumptions without checking the question (external forces can break momentum conservation).
• Mixing up impulses: each object’s impulse is opposite, but total system impulse is zero (if isolated).

5. Exam Tips

• Write your sign convention first: “take rightwards as positive.”
• If the objects stick, replace v₁ and v₂ with one unknown v immediately.
• For elastic collisions, the relative-speed rule is often faster than using kinetic energy directly.
• After solving, do a quick sanity-check:
  • does a heavier object change speed less?
  • do directions (signs) make sense?

6. Worked Examples

7. Mind Stretchers