Lecture 3.3: The Rules of Interaction

In any collision, the total momentum of the system never changes.

Today’s Essential Questions

• What is the Law of Conservation of Momentum?
• How do airbags and crumple zones make collisions safer? (Hint: Use the word ‘impulse’).
• Why is it impossible to eliminate all risk in a vehicle collision?

Connecting to Our Last Investigation

In the crumple zone lab, you saw how extending the collision time reduced the force on the vehicle. This is a perfect, real-world application of the Impulse-Momentum theorem. Today, we expand on that idea with one of the most powerful laws in physics to understand what happens during the interaction itself.

The Law of Conservation of Momentum

This is a fundamental rule of the universe: In an isolated system, the total momentum before an interaction is equal to the total momentum after the interaction.

“Isolated system” means there are no external net forces (like friction or air resistance) changing the system’s total momentum. For the brief, intense moment of a car crash, we can treat the colliding cars as an isolated system.

\(\Sigma p_{initial} = \Sigma p_{final}\) \(p_{1,i} + p_{2,i} = p_{1,f} + p_{2,f}\)

Momentum isn’t lost; it’s just transferred between objects in the system.

Elastic vs. Inelastic Collisions

Collisions can be broadly categorized into two types:

• Elastic: Objects bounce off each other perfectly. Kinetic energy is conserved. (Think billiard balls).
• Inelastic: Objects deform, stick together, or generate heat/sound. Kinetic energy is not conserved.

All real-world vehicle collisions are highly inelastic. Energy is intentionally used to crumple the car rather than being transferred to the occupants.

How Safety Features Really Work

Safety features don’t reduce the total change in momentum ($\Delta p$) in a crash. A car going 50 mph must still come to a stop. Instead, they manipulate the impulse.

From the Impulse-Momentum Theorem: \(F_{net} \Delta t = \Delta p\)

The Goal: To bring the occupant’s momentum to zero, safety features are designed to make the collision time ($\Delta t$) as large as possible. A larger time means the force ($F_{net}$) on the occupant is much smaller and therefore survivable.

The System of Safety

A car’s safety features work together as a coordinated system:

1. Crumple Zone: The front of the car collapses, increasing the time it takes the car to stop. This is the first and longest part of the impulse.
2. Seatbelts: Couple the occupant to the rigid frame of the car, allowing them to “ride down” the collision over the same long time interval.
3. Airbags: Provide a soft surface that increases the time of the occupant’s secondary collision with the car’s interior, reducing the force on the head and chest.

The Engineer’s Trade-off

Why can’t we make a perfectly safe car? Because of engineering trade-offs.

• Too much crumple zone: A car that crumples in a 5 mph parking lot bump would be too expensive to repair.
• Too little crumple zone: A car that’s too rigid stops too quickly, delivering dangerous forces to the occupants.
• Cost vs. Safety: Adding more safety features increases the vehicle’s cost, weight, and complexity.

Engineers must balance the competing criteria of safety, cost, and repairability.

Thinking Lens: Systems and System Models

A safe vehicle is a coordinated system of features (crumple zones, airbags, seatbelts, rigid passenger cell).

Question: Why is it critical for these parts to work together as a system? What would happen, for example, if an airbag deployed but the occupant wasn’t wearing a seatbelt?

Preparing for Our Next Task

You will apply the principles of impulse and conservation of momentum to evaluate the costs and benefits of different airbag designs in our next lab, “Airbag Design Tradeoffs”. You will have to think like an engineer and justify your own design choices.

Summary: Answering Our Questions

• What is the Law of Conservation of Momentum? In an isolated system, the total momentum is constant; it is transferred between objects during a collision but never lost.

• How do airbags and crumple zones make collisions safer? They don’t reduce the total change in momentum, but they increase the time of the impulse, which dramatically reduces the force on the occupants.

• Why is it impossible to eliminate all risk? Engineers must make trade-offs between perfect safety, cost, and everyday functionality. A car that can survive any crash might not be affordable or practical.

Prompt: In 2-3 sentences, explain how a car’s crumple zone saves lives by increasing the time of impact during a collision.