Conservation of momentum and energy

Welcome to the World of Crashes and Smashes!

In this chapter, we are going to look at what happens when things run into each other. Whether it's two billiard balls clicking together or a car bumper-to-bumper in traffic, the physics of collisions is happening! Don't worry if this seems a bit "maths-heavy" at first; we’ll break it down into simple rules that apply to almost everything in the universe.

1. Linear Momentum: The "Moving Power"

Before we talk about collisions, we need to know what momentum is. Think of it as how hard it is to stop a moving object.

The Definition:
Linear momentum \(p\) is the product of an object's mass \(m\) and its velocity \(v\).
\(p = mv\)

Unit: \(kg \cdot m \cdot s^{-1}\)
Important: Momentum is a vector. This means direction matters! If moving to the right is positive (+), then moving to the left must be negative (-).

Analogy: Imagine a heavy truck and a small bicycle both moving at 5 m/s. Which one is harder to stop? The truck! Because it has more mass, it has more momentum.

Quick Review:

• Mass is how much "stuff" is in an object.
• Velocity is speed with a specific direction.
• Momentum is mass in motion.

2. Impulse: The Change in Momentum

How do we change an object's momentum? We apply a force over a certain amount of time. This is called Impulse.

From Newton’s Second Law, we know that Force is the rate of change of momentum:
\(F = \frac{\Delta p}{\Delta t}\)

If we rearrange this, we get Impulse (\(\Delta p\)):
\(\Delta p = F \Delta t\)

The Force-Time Graph

In many collisions (like hitting a tennis ball), the force isn't constant. It starts small, peaks, and then drops. To find the total impulse from a graph of Force against Time, you simply find the area under the graph.

Key Takeaway: Impulse = Change in Momentum = Area under Force-Time graph.

3. The Principle of Conservation of Momentum (PCM)

This is the "Golden Rule" of collisions. It states that:

In a closed system (where no external resultant force acts), the total momentum before a collision is equal to the total momentum after the collision.

In simple terms:
\(Total \ momentum_{initial} = Total \ momentum_{final}\)
\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\)

Did you know? This rule works even if the objects explode apart! As long as there are no outside forces (like friction pulling on the objects from the floor), the "momentum bank account" stays balanced.

Common Mistake to Avoid: Always define a positive direction! If Ball A is moving at 5 m/s to the right and Ball B is moving at 3 m/s to the left, their initial momenta are \(m_A(5)\) and \(m_B(-3)\). Forget the minus sign, and the whole calculation will crash!

4. Elastic vs. Inelastic Collisions

While momentum is always conserved in a closed system, Kinetic Energy (KE) is a different story. Not all collisions "bounce" the same way.

A. Perfectly Elastic Collisions

• Momentum: Conserved.
• Kinetic Energy: Conserved (Total KE before = Total KE after).
• Reality Check: These are rare in our daily lives (usually happening at the subatomic level), but billiard balls come very close.

B. Inelastic Collisions

• Momentum: Conserved.
• Kinetic Energy: NOT conserved. Some energy is converted into heat, sound, or the work done to deform the objects (like a dented car).
• Completely Inelastic: This is a special case where the two objects stick together after colliding and move with the same final velocity.

Analogy: Think of a "Superball" as elastic (it bounces back high) and a lump of wet clay as completely inelastic (it just splats and sticks to the floor).

5. The Special Trick for Elastic Collisions

If you are told a collision is perfectly elastic, there is a very handy shortcut you can use without needing to calculate all the squares in the KE formula (\(\frac{1}{2}mv^2\)).

Relative Speed Rule:
The relative speed of approach is equal to the relative speed of separation.

Formula:
\(u_1 - u_2 = v_2 - v_1\)

What does this mean? If Object A is approaching Object B at a speed of 10 m/s, then after an elastic collision, they will be moving away from each other at exactly 10 m/s.

Step-by-Step for Solving Collision Problems:

1. Draw a diagram: Draw the objects "Before" and "After".
2. Choose a direction: Mark which way is positive (usually right).
3. Write the PCM Equation: \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\).
4. Check for Energy: If it's elastic, use the relative speed equation. If it's inelastic, you may need to calculate the "loss in KE" by comparing \(\frac{1}{2}mv^2\) before and after.

Summary Table: What is Conserved?

Type of Collision | Momentum | Total Energy | Kinetic Energy
Elastic | Conserved | Conserved | Conserved
Inelastic | Conserved | Conserved | NOT Conserved (Lost to heat/sound)
Closed System | Conserved | Conserved | Depends on the collision type

Final Encouragement

Collisions can feel confusing because of the different types of energy and the vector directions. Just remember: Momentum is the loyal friend—it is always there (conserved). Kinetic Energy is the "fair-weather friend"—it might disappear if things get messy! Keep practicing your sign conventions (+ and -), and you'll master this chapter in no time.