Crashing Into Physics: Elastic and Inelastic Collisions

Welcome back! In the previous chapter, we learned that momentum is conserved in any closed system. But what happens to the energy when two objects hit each other? Does it just stay the same, or does some of it "disappear"? In this chapter, we explore the different types of collisions and how to use the laws of physics to predict what happens after the "big bang."

Don't worry if this seems a bit mathematical at first. We are going to break it down into three simple categories so you can easily tell them apart!

1. The Universal Rule: Momentum is King

Before we dive into the differences, there is one thing you must always remember: In a closed system (where no external net force acts), total linear momentum is ALWAYS conserved.

Whether the objects bounce, stick, or explode, the total momentum before the crash equals the total momentum after the crash:

\( \vec{p}_{total, i} = \vec{p}_{total, f} \)

\( m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f} \)

Quick Reminder: Momentum is a vector! If an object is moving to the left, its velocity must be negative in your equations.

2. Elastic Collisions: The Perfect Bounce

An elastic collision is a special type of encounter where the objects bounce off each other so perfectly that no kinetic energy is lost.

Key Characteristics:

  • Momentum is conserved: \( \sum \vec{p}_i = \sum \vec{p}_f \)
  • Kinetic Energy (\( K \)) is conserved: \( \sum K_i = \sum K_f \)
  • The objects do not deform permanently and no heat or sound is generated (in an ideal world).

Real-World Example: Truly elastic collisions are rare in our macroscopic world. However, subatomic particles (like two protons crashing) or "hard" objects like billiard balls or steel spheres are very close to being perfectly elastic.

Key Takeaway: If a problem tells you a collision is elastic, you have two "powers": you can set up a momentum conservation equation AND a kinetic energy conservation equation.

3. Inelastic Collisions: The Real World

In an inelastic collision, the objects bounce off each other, but some of the kinetic energy is converted into other forms of energy, like thermal energy (heat), sound, or the energy used to deform the objects (like a dent in a car).

Key Characteristics:

  • Momentum is conserved: \( \sum \vec{p}_i = \sum \vec{p}_f \) (This is still true!)
  • Kinetic Energy is NOT conserved: \( \sum K_i > \sum K_f \)

Common Mistake: Many students think "inelastic" means momentum isn't conserved. That is a trap! Momentum is always conserved in these collisions; it's only the kinetic energy that changes.

4. Perfectly Inelastic Collisions: The "Sticky" Situation

A perfectly inelastic collision (sometimes called a completely inelastic collision) is the easiest to identify. This happens when the two objects stick together after they collide and move as a single unit.

The Math of Sticking Together:

Since the two objects become one mass, they share the same final velocity (\( v_f \)). The conservation of momentum equation looks like this:

\( m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f \)

Key Characteristics:

  • This type of collision results in the maximum possible loss of kinetic energy for the system (while still obeying momentum conservation).
  • Example: A piece of gum thrown at a moving skateboard, or a railroad car coupling with another.
Did you know?

In a perfectly inelastic collision, the kinetic energy "lost" isn't destroyed—it just changes into "internal energy" (like heat or the energy required to make the objects stick). Physics never loses energy; it just hides it!

5. Energy Consequences: Doing the Math

On the AP Exam, you might be asked to justify what kind of collision occurred. To do this, you usually need to calculate the total kinetic energy before and after.

Step-by-Step Energy Analysis:

1. Calculate initial Kinetic Energy: \( \sum K_i = \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 \)

2. Calculate final Kinetic Energy: \( \sum K_f = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \)

3. Compare:
    - If \( \sum K_i = \sum K_f \), it is Elastic.
    - If \( \sum K_i > \sum K_f \), it is Inelastic.

The "Explosion" Scenario: If the kinetic energy increases (\( \sum K_i < \sum K_f \)), this is usually an "explosion" where potential energy (like a spring or chemical propellant) was converted into kinetic energy. Momentum is still conserved!

6. Collisions in Two Dimensions (2D)

Sometimes objects don't hit head-on; they glance off each other at angles (like a trick shot in pool). For AP Physics 1, you only need a semiquantitative understanding of this.

The Secret: Treat the x-direction and y-direction separately!
- The total momentum in the x-direction is conserved: \( \sum p_{ix} = \sum p_{fx} \)
- The total momentum in the y-direction is conserved: \( \sum p_{iy} = \sum p_{fy} \)

If Object A is moving right and hits Object B (at rest), and after the collision Object A moves "up and right," Object B must move "down and right" to cancel out the upward momentum. The total y-momentum must stay zero!

Quick Review Box

Momentum: Always conserved in all collision types (\( \sum \vec{p}_i = \sum \vec{p}_f \)).

Elastic: Kinetic energy is conserved. Objects bounce perfectly.

Inelastic: Kinetic energy is lost to heat/sound/damage. Objects bounce.

Perfectly Inelastic: Kinetic energy is lost. Objects stick together (\( v_{1f} = v_{2f} \)).

Pro Tip: Always check your signs! If an object changes direction, its velocity sign must change from positive to negative (or vice versa).