Perfectly inelastic collision in AP Physics C: Mechanics

A perfectly inelastic collision is a collision in which the objects stick together and move with one shared final velocity afterward. Momentum is always conserved, but the collision loses the maximum possible kinetic energy consistent with momentum conservation.

What is perfectly inelastic collision?

A perfectly inelastic collision is the "stick together" collision. Two objects hit, lock onto each other, and move off as a single combined object with one final velocity. That sticking condition is the whole definition. If the problem says a bullet embeds in a block, a railroad car couples to another car, or clay splats onto a cart, you're looking at a perfectly inelastic collision.

Here's the part that trips people up. Momentum is still conserved, because the only forces during the collision are internal to the system. Kinetic energy is NOT conserved. In fact, a perfectly inelastic collision destroys the most kinetic energy any collision can while still conserving momentum. The "lost" energy goes into deformation, heat, and sound. The math is the cleanest of any collision type, since there's only one unknown final velocity: m₁v₁ + m₂v₂ = (m₁ + m₂)v_f. Solve for v_f, then compare kinetic energy before and after to find how much was converted to other forms.

Why perfectly inelastic collision matters in AP® Physics C: Mechanics

This term lives in Topic 4.4 (Elastic and Inelastic Collisions) in the Linear Momentum unit of AP Physics C: Mechanics. It's the testing ground for one of the biggest ideas in the course, that momentum conservation and energy conservation are separate principles that apply under separate conditions. Perfectly inelastic collisions force you to use momentum conservation while explicitly refusing to let you use kinetic energy conservation. Mastering when each law applies (and when it doesn't) is exactly the kind of reasoning the exam rewards, especially in multi-stage problems like the ballistic pendulum where you have to switch principles mid-problem.

How perfectly inelastic collision connects across the course

Conservation of Linear Momentum (Unit 4)

Momentum conservation is the only tool that survives a perfectly inelastic collision. With no external force on the system, total momentum before equals total momentum after, even though kinetic energy drops. Every perfectly inelastic problem starts with m₁v₁ + m₂v₂ = (m₁ + m₂)v_f.

Elastic Collisions (Unit 4)

These are the two ends of the collision spectrum. Elastic collisions conserve kinetic energy; perfectly inelastic collisions lose the maximum amount possible. Real collisions fall somewhere in between, which is why the exam loves asking you to classify a collision from given data.

Conservation of Energy (Unit 3)

The classic ballistic pendulum stitches the two units together. You use momentum conservation during the bullet-block collision (perfectly inelastic), then switch to energy conservation as the block swings up. Using energy conservation during the collision itself is the single most common error on this problem type.

Center of Mass (Unit 4)

After a perfectly inelastic collision, the combined object moves at the velocity of the system's center of mass. That velocity never changed during the collision, which is a slick way to see why total kinetic energy can't drop to zero unless the total momentum was zero to begin with.

Is perfectly inelastic collision on the AP® Physics C: Mechanics exam?

Multiple-choice questions usually hand you masses and an initial velocity and ask for the final shared velocity, the fraction of kinetic energy lost, or the classification of the interaction. For example, a 2.0 kg object at 6.0 m/s sticking to a stationary 3.0 kg object loses 60% of its kinetic energy, and a neutron embedding in a carbon nucleus 12 times its mass loses 12/13 of its kinetic energy. A useful shortcut when the target starts at rest is that the fraction of kinetic energy lost equals M/(m + M), the target's share of the total mass. Questions also test the edge case where two identical objects with equal and opposite momenta stick together; total momentum is zero, so 100% of the kinetic energy can be lost. On free-response, the term shows up inside multi-step setups like the ballistic pendulum, where the graders are checking whether you apply momentum conservation to the collision and energy conservation to the swing, and never the reverse.

Perfectly inelastic collision vs Inelastic collision (general)

Every collision that loses kinetic energy is inelastic, but "perfectly" inelastic is a specific subset where the objects stick together and share one final velocity. A regular inelastic collision can have the objects bounce apart with different velocities while still losing some kinetic energy. Perfectly inelastic is the extreme case, losing the maximum kinetic energy that momentum conservation allows. If the problem doesn't say the objects stick (or embed, couple, or merge), don't assume the final velocities are equal.

Key things to remember about perfectly inelastic collision

• In a perfectly inelastic collision, the objects stick together and move with one shared final velocity, so there's only one unknown: v_f = (m₁v₁ + m₂v₂)/(m₁ + m₂).

• Momentum is conserved in a perfectly inelastic collision, but kinetic energy is not, and you should never set KE_initial equal to KE_final.

• A perfectly inelastic collision loses the maximum kinetic energy possible while still conserving momentum; the lost energy becomes heat, sound, and deformation.

• When a moving object sticks to a stationary one, the fraction of kinetic energy lost equals the stationary object's mass divided by the total mass, M/(m + M).

• If two objects with equal and opposite momenta stick together, the final velocity is zero and 100% of the kinetic energy is lost.

• In ballistic pendulum problems, use momentum conservation for the embedding collision and energy conservation for the swing afterward, in that order.

Frequently asked questions about perfectly inelastic collision

What is a perfectly inelastic collision in AP Physics C?

It's a collision where the objects stick together and move with the same final velocity afterward. Momentum is conserved, kinetic energy is not, and the collision loses the maximum kinetic energy consistent with momentum conservation.

Is momentum conserved in a perfectly inelastic collision?

Yes. As long as there's no net external force on the system, total momentum is conserved in every collision, including perfectly inelastic ones. It's kinetic energy that isn't conserved.

Is all kinetic energy lost in a perfectly inelastic collision?

No, not usually. The combined object still moves at the center-of-mass velocity, so it keeps some kinetic energy. The only way to lose 100% is if the total momentum is zero, like two identical objects colliding head-on at equal speeds.

What's the difference between an inelastic and a perfectly inelastic collision?

An inelastic collision is any collision that loses kinetic energy; the objects can still bounce apart. A perfectly inelastic collision is the special case where they stick together and share one final velocity, which maximizes the kinetic energy lost.

How do you solve a ballistic pendulum problem?

Split it into two stages. First, use momentum conservation for the bullet embedding in the block, since that collision is perfectly inelastic: mv = (m + M)v_f. Then use energy conservation for the swing, setting ½(m + M)v_f² equal to (m + M)gh. Mixing up which law goes with which stage is the classic mistake.