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Elastic & Inelastic
Collisions

Master the physics of collisions: when kinetic energy is conserved, when it isn't, and how momentum ties it all together.

AP Physics 1

What Is a Collision?

A collision occurs when two or more objects exert forces on each other over a brief time interval. During this interaction, external forces (like gravity or friction) are typically negligible compared to the enormous internal forces between the objects.

BEFORE \(m_1\) \(m_2\) \(v_1\) AFTER \(m_1\) \(v'_1\) \(m_2\) \(v'_2\)

Momentum Conservation

Linear momentum is defined as \( \vec{p} = m\vec{v} \). In any collision between two objects where external forces are negligible:

This equation holds for every type of collision: elastic, inelastic, and perfectly inelastic. It comes directly from Newton's Third Law: the forces objects exert on each other are equal and opposite, so the total impulse on the system is zero.

Vector Quantity
Momentum has direction: assign \( + \) and \( - \) for opposite directions.
System Scope
Conservation applies to the system, not to each object individually.

Kinetic Energy in Collisions

While momentum is always conserved, kinetic energy may or may not be. The kinetic energy of an object is:

During a collision, kinetic energy can be converted into heat, sound, deformation, or other forms of energy. Whether this conversion happens determines the type of collision.

Elastic
KE is fully conserved: total KE before = total KE after.
Inelastic
Some KE is lost to deformation, heat, or sound.
Perfectly Inelastic
Objects stick together: maximum KE loss.

Elastic Collisions: Two Conservation Laws

In an elastic collision, both momentum and kinetic energy are conserved. This gives us two simultaneous equations:

Perfectly Inelastic Collisions

In a perfectly inelastic collision, the two objects stick together and move as one mass after the collision. Only momentum is conserved:

BEFORE \(m_1\) \(v_1\) \(m_2\) at rest AFTER \(m_1\!+\!m_2\) stuck \(v_f\)

This type loses the maximum amount of kinetic energy possible while still conserving momentum. Examples: a football tackle, a ball of clay hitting a wall.

Elastic vs. Inelastic: Key Distinction

Property Elastic Inelastic Perfectly Inelastic
Momentum conserved? ✓ Yes ✓ Yes ✓ Yes
KE conserved? ✓ Yes ✕ No ✕ No
Objects after? Bounce apart Bounce (some KE lost) Stick together
KE lost to Nothing Sound, heat, deformation Maximum loss
Real example Billiard balls (approx.) Car crash (cars separate) Catching a ball

A cart collision

This source example asks about sticking. The graphic also shows the elastic outcome for the same incoming cart masses and speed.

A 4.0 kg cart moving at 6.0 m/s to the right collides with a 2.0 kg cart initially at rest on a frictionless track. The two carts lock together after the collision. Find (a) the final velocity and (b) the fraction of kinetic energy lost.

Show answer and explanation

Step 2 — Classify the Collision: The carts lock together, so this is a perfectly inelastic collision. We use: m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)vf. Step 3 — Apply Conservation of Momentum: Substituting: (4.0 kg)(6.0 m/s) + (2.0 kg)(0 m/s) = (4.0 + 2.0 kg) × vf → 24 kg·m/s = 6.0 kg × vf. vf = 4.0 m/s to the right Step 4 — Calculate Initial Kinetic Energy: KEᵢ = ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½(4.0)(6.0)² + 0 = 72 J. KEᵢ = 72 J Step 5 — Calculate Final Kinetic Energy: KEf = ½(m₁ + m₂)vf² = ½(6.0)(4.0)² = 48 J. KEf = 48 J Step 6 — Determine Fraction of KE Lost: ΔKE = 72 − 48 = 24 J. Fraction lost = ΔKE / KEᵢ = 24 / 72 = 1/3 ≈ 0.33. One-third of the kinetic energy was converted into internal energy (heat, sound, deformation). Notice that in any perfectly inelastic collision where one object is initially at rest, the fraction of KE lost equals m₂ / (m₁ + m₂), which here is 2/6 = 1/3. Fraction lost = 1/3 (about 33%)

AP Physics 1: a 4 kg cart at 6 m/s strikes a stationary 2 kg cart. An elastic collision gives velocities 2 and 8 m/s; sticking gives both 4 m/s. Momentum remains 24 kg·m/s. The C companion uses different masses.
AP Physics 1: a 4 kg cart at 6 m/s strikes a stationary 2 kg cart. An elastic collision gives velocities 2 and 8 m/s; sticking gives both 4 m/s. Momentum remains 24 kg·m/s. The C companion uses different masses.

Why Momentum Is Always Conserved

1
Newton's Third Law: When object 1 pushes on object 2 with force \( \vec{F} \), object 2 pushes back with \( -\vec{F} \).
2
Equal impulses: Both forces act for the same time \( \Delta t \), so \( \vec{J}_1 = -\vec{J}_2 \).
3
Impulse = change in momentum: \( \Delta \vec{p}_1 = -\Delta \vec{p}_2 \), meaning what one gains, the other loses.
4
Total change = zero: \( \Delta \vec{p}_1 + \Delta \vec{p}_2 = 0 \), so \( \vec{p}_{\text{total}} \) is constant.

How to Solve Collision Problems

1
Identify the system: List the objects involved and confirm external forces are negligible during the collision.
2
Choose a direction: Define positive \( (+) \) for one direction. Assign velocities with correct signs.
3
Apply momentum conservation: Write \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \).
4
Determine the collision type: If elastic, add the KE equation (or relative velocity condition). If perfectly inelastic, set \( v_{1f} = v_{2f} = v_f \).
5
Solve: Use algebra to find unknowns. Check: does your answer make physical sense (direction, magnitude)?

Example 1: Elastic Collision

A \( 2\,\text{kg} \) ball moving at \( 5\,\text{m/s} \) collides elastically with a \( 3\,\text{kg} \) ball at rest. Find the final velocities of both balls.

\( 2\,\text{kg} \) \( 5\,\text{m/s} \) \( 3\,\text{kg} \) at rest AFTER: \( 2\,\text{kg} \) \( -1\,\text{m/s} \) \( 3\,\text{kg} \) \( 4\,\text{m/s} \)

Example 2: Perfectly Inelastic

A \( 4\,\text{kg} \) cart moving at \( 6\,\text{m/s} \) collides with and sticks to a \( 2\,\text{kg} \) cart moving at \( -3\,\text{m/s} \) (opposite direction). Find the final velocity and the kinetic energy lost.

Example 3: Ballistic Pendulum

A \( 0.01\,\text{kg} \) bullet embeds in a \( 2\,\text{kg} \) block hanging from a string. The block+bullet rises \( 0.45\,\text{m} \). Find the bullet's speed.

\( 2\,\text{kg} \) \( 0.01\,\text{kg} \) \(v = ?\) → \( 2.01\,\text{kg} \) \(h = 0.45\,\text{m}\) BEFORE AFTER

The impact that becomes a pendulum swing

A level capture block hangs on parallel cords inside a protective enclosure, with an embedded pellet visible in a conceptual window.Treat capture and swing as separate stages. Horizontal momentum is conserved during the short impact; mechanical energy is conserved during the subsequent ideal swing.

The impact that becomes a pendulum swing

The impact that becomes a pendulum swing. Treat capture and swing as separate stages. Horizontal momentum is conserved during the short impact; mechanical energy is conserved during the subsequent ideal swing.Treat capture and swing as separate stages. Horizontal momentum is conserved during the short impact; mechanical energy is conserved during the subsequent ideal swing.

Special Elastic Cases

Equal Masses
If \( m_1 = m_2 \) and object 2 is at rest: object 1 stops, object 2 moves with object 1's initial velocity. They "exchange" velocities.
Heavy Hits Light
If \( m_1 \gg m_2 \): the heavy object barely changes speed, while the light object flies off at nearly twice the heavy object's speed.
Light Hits Heavy
If \( m_1 \ll m_2 \): the light object bounces back at nearly the same speed, and the heavy object barely moves. (Think: ball bouncing off a wall.)

Quantifying Energy Loss

For a perfectly inelastic collision where object 2 is initially at rest, the fraction of kinetic energy lost has a clean formula:

Equal Masses
\( m_1 = m_2 \): exactly 50% of KE is lost.
Heavy Target
\( m_2 \gg m_1 \): nearly all KE is lost (bullet into block).

Common Mistakes

✕ Forgetting Signs
Velocity is a vector. If two objects move in opposite directions, one velocity must be negative. Forgetting this gives wrong momentum totals.
✕ Assuming KE is Always Conserved
Only elastic collisions conserve KE. In AP problems, if objects deform or stick, you cannot use the KE conservation equation.
✕ Confusing Elastic with Perfectly Inelastic
"Inelastic" does not mean objects stick. Perfectly inelastic means they stick. Regular inelastic means some KE is lost but objects separate.

Summary of Collision Equations

Equation Elastic Inelastic Perf. Inelastic
\( \Sigma p_i = \Sigma p_f \) ✓ ✓ ✓
\( \Sigma KE_i = \Sigma KE_f \) ✓ ✕ ✕
Relative velocity condition ✓ ✕ ✕
Objects stick? ✕ ✕ ✓
# of unknowns solvable 2 (two equations) 1 (one equation) 1 (one equation)

Question 1 of 5

In all collisions (elastic, inelastic, and perfectly inelastic), which quantity is always conserved?

Question 2 of 5

A \( 3\,\text{kg} \) object moving at \( 4\,\text{m/s} \) collides with and sticks to a \( 1\,\text{kg} \) object at rest. What is the final velocity of the combined object?

Question 3 of 5

Two objects of equal mass undergo a head-on elastic collision. Object A is moving at \( 6\,\text{m/s} \) and object B is at rest. What happens after the collision?

Question 4 of 5

A \( 5\,\text{kg} \) ball moving at \( 8\,\text{m/s} \) collides with a \( 5\,\text{kg} \) ball at rest. After the collision, the first ball moves at \( 2\,\text{m/s} \) in the original direction. How much kinetic energy was lost?

Question 5 of 5

Which of the following collisions loses the greatest fraction of the system's initial kinetic energy?