Topic 4.4: Elastic and Inelastic Collisions

What Topic 4.4 requires

Topic 4.4 has one learning objective, 4.4.A: describe whether an interaction between objects is elastic or inelastic. Five essential knowledge statements sit under it, and every one of them is about kinetic energy.

• 4.4.A.1 An elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system.
• 4.4.A.2 In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies.
• 4.4.A.3 An inelastic collision between objects is one in which the total kinetic energy of the system decreases.
• 4.4.A.4 In an inelastic collision, some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy.
• 4.4.A.5 In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision.

Notice what is missing. Not one statement in Topic 4.4 mentions momentum, because momentum was settled in Topic 4.3 and applies to all three cases without modification. Classification is purely an energy question.

The CED's suggested skills here are 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.A, derive a symbolic expression from known quantities; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim. Unit 4 is weighted at 10 to 15 percent of the multiple-choice section.

Momentum is conserved in both, kinetic energy only in elastic

Here are the direct answers to the questions this topic gets asked most.

• Is momentum conserved in an elastic collision? Yes.
• Is momentum conserved in an inelastic collision? Yes.
• Is momentum conserved in a perfectly inelastic collision? Yes.
• In which type of collision is kinetic energy conserved? Elastic collisions, and only elastic collisions.

One condition sits behind all four answers, and it comes from 4.3.B.2 rather than from this topic: a system's total momentum is constant whenever the net external force on it is zero. That condition is about what crosses the system boundary, not about whether the objects bounce, stick, crumple, or heat up. Two carts on a frictionless track have no net external force on them whether they carry magnetic bumpers or sticky pads, so both collisions keep the same total momentum.

Kinetic energy has no such protection, because it is one category of energy rather than a separately conserved quantity. Total energy is always conserved, and a collision is free to move energy out of the kinetic category into thermal energy, sound, and permanent deformation. What sets the elastic case apart is that nothing moves out: the objects deform and spring back completely, and 4.4.A.1 defines that outcome as the system's initial kinetic energy being equal to its final kinetic energy.

Elastic collisions (4.4.A.1 and 4.4.A.2)

An elastic collision is one where the system's total kinetic energy comes out the same as it went in:

Statement 4.4.A.2 adds the qualification students most often miss: in an elastic collision the final kinetic energies of each of the objects may be different from their initial kinetic energies. Elastic means the total is preserved, not that each object keeps what it brought. Energy moves freely between the two objects and only the sum is fixed. In the first worked example below, one cart ends with about 6 percent of the kinetic energy it started with, and the collision is still perfectly elastic.

Two standard results are worth recognizing on sight, both for a one-dimensional elastic collision with a target that starts at rest.

• Equal masses. The incoming object stops dead and the target leaves with the incoming velocity. The two swap velocities, which is what a Newton's cradle demonstrates.
• Light object, much heavier target. The light object rebounds backward at nearly its original speed while the heavy one barely moves, the way a ball bounces off a wall.

At human scale the elastic case is an idealization that laboratory equipment is built to approach: magnetic and spring bumpers on carts, hardened steel spheres, air-track gliders. It is treated as exact in the kinetic theory of gases, where molecular collisions are modeled as losing nothing. One practical note: $K$ depends on $v^2$, so its value never depends on which way an object is travelling. Signs matter enormously in the momentum equation and not at all in the energy one.

Inelastic collisions (4.4.A.3 and 4.4.A.4)

An inelastic collision is one in which the total kinetic energy of the system decreases (4.4.A.3). Statement 4.4.A.4 says where it went: some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy.

Nonconservative is the operative word, and it points straight back to potential energy in Unit 3. A conservative force stores energy in a potential energy that gives it back in full; a nonconservative force does not. When two cars crumple, the metal deforms permanently and the work spent bending it never returns as motion. The energy is still in the universe as thermal energy, sound, and the internal energy of bent steel, so total energy is conserved and only the kinetic share drops.

Most real collisions are inelastic. Two carts that bounce apart at reduced relative speed, a basketball that returns to less than the height it was dropped from, a tennis ball off a racket: all inelastic, and none of them perfectly inelastic, because in each case the objects separate.

To classify a collision this way you have to compute both totals. There is no shortcut from the description, and a tidy-looking setup earns no assumption: elastic is a finding, not a first impression.

Perfectly inelastic collisions lose the most, not all (4.4.A.5)

Statement 4.4.A.5 defines the extreme case: in a perfectly inelastic collision the objects stick together and move with the same velocity after the collision. Because there is now only one final velocity, momentum conservation on its own finishes the problem:

This is the case that gets misremembered. A perfectly inelastic collision loses the maximum kinetic energy that conservation of momentum permits, which is almost never all of it. The combined object usually keeps moving, and anything moving has kinetic energy.

The reason is two lines of algebra. The total momentum $p$ is fixed by conservation, and afterward the whole mass moves together at one velocity, so

Both $p$ and $M$ are settled before the collision even happens, so this much kinetic energy survives no matter what the objects do to each other. Any other momentum-conserving outcome adds to it: split each object's velocity into the shared center-of-mass velocity plus a velocity relative to the center of mass, and the total becomes $K = \frac{p^2}{2M} + \sum \frac{1}{2} m_i u_i^2$. That second sum is a sum of squares, so it is never negative, and it is zero only when every object moves at $v_{\text{cm}}$, which is exactly what sticking together means. So $p^2/2M$ is the floor. It equals zero only when the total momentum is zero, which happens only when the two objects arrive with equal and opposite momenta. That head-on case is the one perfectly inelastic collision that brings everything to a stop.

In the language of Topic 4.3, that second term is the whole story: the kinetic energy of the center-of-mass motion is locked in by conservation of momentum and cannot be touched. Only the kinetic energy of motion relative to the center of mass is available to be lost, and sticking together is precisely what removes all of that. Perfectly inelastic is the floor, not a special zero.

The words that signal this case in a problem are stick, couple, latch, embed, lodge in, and move off together.

The three cases side by side

Every row reads constant in the momentum column, on the standing condition that the net external force on the system is zero. Elastic and inelastic differ only in the third column, which is the whole content of learning objective 4.4.A.

A fourth case exists that Topic 4.4 does not name: an interaction that increases the system's kinetic energy by releasing stored energy. That is the explosion of 4.1.A.3.iii, and a compressed spring released between two carts is the lab version. The total momentum is constant there too, which is one more reminder that the momentum column never depends on what the energy column is doing.

Deciding from data (skills 3.A and 1.B)

Suggested skill 3.A asks you to create experimental procedures appropriate for a given scientific question, and was this collision elastic? is a question the exam's lab-based free-response can put to you directly. One of the CED's sample activities for this topic has students run a nonstick collision between two carts of different masses, film it from above with a meter stick beside the track, and use a frame-by-frame review app to find each cart's initial and final speed, then decide whether momentum was conserved and whether the collision was elastic.

The analysis is a fixed four-step routine.

1. Measure each object's mass and its velocity before and after, with a sign attached to every velocity.
2. Check momentum first. Compare $m_1v_{1i} + m_2v_{2i}$ with $m_1v_{1f} + m_2v_{2f}$. If they disagree by more than your uncertainty, the velocity data or the system choice is wrong and no conclusion about energy is safe yet.
3. Compute the kinetic energy totals using $K = \frac{1}{2}mv^2$ for each object. Every term comes out positive, since $v$ appears squared.
4. Compare the totals. Equal means elastic, smaller after means inelastic, and one shared final velocity means perfectly inelastic.

Running step 2 before step 3 is what saves you. Momentum is the check on your measurements; kinetic energy is the classification.

A clean way to report step 4 is the fraction $K_f/K_i$, which is 1 for an elastic collision and less than 1 for an inelastic one. For a perfectly inelastic collision with a target that starts at rest, that fraction has an exact form, $m_1/(m_1+m_2)$, so a heavier target means a larger fractional loss. Skill 2.C, comparing physical quantities between two or more scenarios, is exactly what that comparison scores. Run scenarios through the momentum collision calculator to watch the fraction move with the mass ratio.

How the exam tests 4.4

Elastic and inelastic show up in three recognizable shapes.

Classification questions. You are handed velocities before and after and asked which category applies, or handed the category and asked what follows from it. Skill 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim, is what is being scored.

Perfectly inelastic calculations. These are where the quantitative questions cluster, because sticking together removes an unknown and leaves one equation with one answer.

Two-phase problems. A collision followed by an energy step: a block with a dart embedded in it slides into a spring, or swings upward on a string. Treat the phases separately. Momentum conservation carries you through the collision, then conservation of energy carries you after it. Kinetic energy does not survive the collision step unless the problem states the collision is elastic. The conservation of momentum guide works a full two-phase problem end to end.

One thing the exam will not do is ask you to solve a general elastic collision for two unknown final velocities. The Topic 4.3 boundary statement excludes questions requiring the solution of simultaneous equations from AP Physics 1, and a two-unknown elastic collision needs exactly that. You can still be asked to set both equations up and to reason about how the outcome shifts when a mass or a speed changes. Nothing on the AP Physics 1 equation sheet gives elastic collision final velocities, which is the sheet quietly agreeing with the same boundary.

For the full problem-solving procedure across every collision type, work through the conservation of momentum guide, and see the Unit 4 overview for how this topic sits alongside the other three.