AP Physics C: Mechanics · Unit 4: Linear Momentum · Lesson 4.4

Deep Dive: Elastic and Inelastic Collisions

🔬 Deep Dive
This is your textbook for this topic. Take your time. Read it more than once.
Back to Lesson 4.4
4.4.A.1Concept

Elastic Collisions

An elastic collision is one in which the system's total kinetic energy is exactly the same before and after the collision:

K(i) = K(f)
🔑Momentum is always conserved in an isolated collision, elastic or not — but kinetic energy conservation is the special, extra condition that specifically defines "elastic." Real-world collisions only ever approximate this — billiard balls and hard steel spheres come close, but true elastic collisions are an idealization.
4.4.A.24.4.A.34.4.A.4Concept

Inelastic Collisions

An inelastic collision is one in which the system's total kinetic energy decreases. Individual objects' final kinetic energies may end up completely different from their initial ones — momentum still balances perfectly, but energy doesn't.

Kinetic energy in →🔥 heat🔊 sound🔧 deformationNot destroyed — converted by nonconservative forces into other forms.
⚠️That "missing" kinetic energy isn't destroyed — energy is always conserved overall (Lesson 3.4). It's transformed by nonconservative forces into other forms: heat from friction during impact, sound from the collision itself, or permanent deformation of the colliding objects (a crumpled car fender, for instance).

Explore the full spectrum from elastic to maximally inelastic directly below.

A 3 kg cart moving at 6 m/s strikes a stationary 5 kg cart. Drag cart 1's final velocity across the full physical range — momentum conservation determines cart 2's velocity automatically, and the classification updates live.

v₁f-0.75 m/s
Inelastic
v₂f (from momentum conservation) = 4.05 m/sK(initial) = 54.0 J  ·  K(final) = 41.9 J (78% of initial)

Slide all the way to one end and you get the elastic collision (100% KE retained); slide to the other end and cart 1 exactly matches cart 2's velocity — perfectly inelastic. Every point in between is a valid inelastic outcome, all sharing the same conserved momentum.

4.4.A.5Math

Perfectly Inelastic Collisions

A perfectly inelastic collision is the most extreme case of an inelastic collision: the objects stick together entirely, moving off afterward with one shared final velocity.

Since the two objects effectively merge into a single object of combined mass, conservation of momentum takes a particularly clean form — exactly matching this unit's own header art:

m₁v⃗₁ᵢ + m₂v⃗₂ᵢ = (m₁+m₂)v⃗f

Explore this directly below.

The "Perfectly Inelastic" scenario from this unit's header art. Adjust both carts — watch the shared final velocity and the kinetic energy that doesn't survive the collision.

m₁4 kg
v₁8 m/s
m₂6 kg
v₂0 m/s
vf = (m₁v₁+m₂v₂)/(m₁+m₂) = 3.20 m/sK(initial) = 128.0 J  ·  K(final) = 51.2 JKE lost = 76.8 J (60% of initial)

A perfectly inelastic collision always loses some kinetic energy (except in the trivial case where both objects already share the same velocity) — sticking together takes energy that never comes back as motion.

ExampleGuided Example — Classifying a Collision

A 2 kg cart moving at 10 m/s collides with a stationary 2 kg cart. After the collision, both carts move together at 5 m/s. Classify this collision.

Step 1 — Check whether momentum is conserved
p(i) = (2)(10) + (2)(0) = 20 kg·m/s. p(f) = (2+2)(5) = 20 kg·m/s. Momentum checks out — consistent with any isolated collision.
ExampleWorked Example — The Arrow and the Pumpkin

An arrow of mass 0.05 kg moving at 40 m/s strikes a stationary 3 kg pumpkin hanging from a string, embedding itself completely (a perfectly inelastic collision). Find the pumpkin-and-arrow system's velocity immediately after impact, and the fraction of kinetic energy lost.

← Back to Lesson 4.4That's a wrap on Unit 4! Unit 5 shifts into Torque and Rotational Dynamics — force and Newton's second law, reimagined for spinning objects.